## 1. Concept of Matrices and Types of Matrices

Chapter-3

Matrices

Concept of Matrices and Types of Matrices

Matrices

Definition

A matrix is a rectangular array of numbers or functions or objects  arranged in row and column.

The numbers or functions or objects in the array are called the entities or members or the elements of the matrix. The horizontal array of elements in the matrix is called rows, and the vertical array of elements are called the columns. If a matrix has m rows and n columns, then it is known as the matrix of order ‘m x n’ or ‘m by n’.

Symbol: ( ) or [ ]

i.e., A = [aij]m×n  means m rows and n columns.

Note: If A = [aij] n×n  is a square matrix of order n, then elements a11, a22, a33,…, ann is said to constitute the diagonal of the matrix A. In general, A = [aij]m×m is a diagonal matrix, if aij = 0, when i ≠ j.

## Types of Matrices:

1. Column matrix
2. Row matrix
3. Square matrix
4. Diagonal matrix
5. Scalar matrix
6. Identity matrix
7. Zero matrix

Equality of Matrices

Two matrices are said to be equal if-

(i) The order of both the matrices are the same

(ii) Each element of one matrix is equal to the corresponding element of the other matrix

i.e.,

## 1. Concept of Determinant and Properties of Determinants

Chapter-4

Determinants

A determinant is a square array of numbers (written within a pair of vertical lines) which represents a certain sum of products. Below is an example of a 3 × 3 determinant (it has 3 rows and 3 columns). The result of multiplying out, then simplifying the elements of a determinant is a single number (a scalar quantity).

Concept of Determinant and Properties of Determinants:

A  system of linear equations like

a1 x + b1 y = c1

a2 x + b2 y = c2

Now, this system of equations has a unique solution or not, is determined by the number.

a1 b2 – a2 b1 ( Recall that if

or, a1 b2 – a2 b1 0, then the system of linear

equations has a unique solution). The number  a1 b2 – a2 b1 which determines uniqueness of solution is associated with the matrix  A = a1a2b1b

and is called the determinant of A or det A.

Determinants have wide applications in Engineering, Science, Economics, Social Science, etc.

Determinant :

The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix A is denoted det(A), det A, or |A|.

In the case of a 2 × 2 matrix the determinant can be defined as

To every square matrix A = [aij] of order n, we can associate a number (real or complex) called determinant of the square matrix A, where a = (i, j)th element of A. This may be thought of as a function which associates each square matrix with a unique number (real or complex).

If M is the set of square matrices, K is the set of numbers (real or complex) and f : M → K is defined by f (A) = k, where A M and k K, then f (A) is called the determinant of A. It is also denoted by | A | or det A or Δ.

Properties of Determinants

Let us check the below seven properties of determinant in detail. The working principle and the formulas, explanation of each of the properties is also presented below.

1. Interchange Property

The value of a determinant remains unchanged if the rows or the columns of a determinant are interchanged.

Det(A) = Det(A')

It follows from this property that if the rows and columns of the matrix are interchanged, then the transpose of the matrix is obtained and the determinant value and the determinant of the transpose are equal.

2. Sign Property

The sign of the value of determinant changes if any two rows or any two columns are interchanged.

Det(A) = -Det(B)

The value of the determinant only changes the sign if the row or the column is swapped once. In the above matrix A, the second row has been swapped with the third row to obtain matrix B, and we have Det(A) = -Det(B). If the value of the determinant is D, and the rows or columns are swapped n times, then the new value of the determinant is (-1)nD.

3. Zero Property

The value of a determinant is equal to zero if any two rows or any two columns have the same elements.

Here the elements of the first row and the second row are identical. Hence the value of the determinant is equal to zero.

Det(A) = 0

### 4. Scalar Multiplication Property

The value of the determining becomes k times the earlier value of the determinant if each of the elements of a particular row or column is multiplied with a constant k.

Det(B) = k× Det(B)

The elements of the first row are multiplied with a constant k, and the determinant value is also multiplied with the constant k. This property helps in taking a common factor from each row or a column of the determinant. Also if the corresponding elements of any two rows or columns are equal then the value of the determinant is equal to zero.

5. Sum Property

If a few elements of a row or column are expressed as a sum of terms, then the determinant can be expressed as a sum of two or more determinants.

The elements of the first row represent the sum of terms, which can be split into two different determinants. Further, the new determinants also have the same second and third row, as the earlier determinant.

6. Property Of Invariance

If each element of a row and column of a determinant is added with the equimultiples of the elements of another row or column of a determinant, then the value of the determinant remains unchanged. This can be expressed in the form of a formula as Ri→Ri+kRj, or Ci→Ci+kCj.

The elements of the first row of matrix A have been replaced with the sum of the elements of the first row, and the third row multiplied with a constant, to obtain the new matrix B. Here, after this operation also, the determinant A is equal to determinant B.

7. Triangular Property

If the elements above or below the main diagonal are equal to zero, then the value of the determinant is equal to the product of the elements of the diagonal matrix.

8. Factor Property:

If a determinant Δ becomes zero when we put

x=α,

Then
(x-α)

is a factor of Δ.

### 9. Determinant of cofactor matrix:

where Cij denotes the cofactor of three element aij in Δ .

### Question : Using properties of determinants, prove that

Solution:

By using invariance and scalar multiple property of determinant we can prove the given problem.

## Operations on Matrices

The  addition of matrices, multiplication of a matrix by a scalar, difference and multiplication of matrices. However, division of matrices is not possible in a direct way.

If   A=[aij]  and   B=[bij]

are two matrices of the same order, say m × n, then the sum of the two matrices A and B is defined as a matrix C=[cij]m×n

, where cij=aij+bij

, for all possible values of i and j.

i.e. C = A + B

are two matrices of order 3 x 2 such that the sum of these two matrices is given by:

C = A + B

Therefore,

That means the sum or addition of two matrices is a matrix obtained by adding the corresponding elements of the given two matrices. Also, it is essential to note that the two matrices have to be of the same order.

Here, the order of the matrix A is 2 x 3 and the order of B is 2 x 3 are the same. So, we can add the given two matrices by adding the corresponding elements.

Example: Suppose

, then the addition of A and B is not possible since the order of matrix A is 2 x 2 and the order of B is 2 x 3, i.e. the order of these matrices is not equal.

## Properties of Addition of Matrices

[1] Commutative Law: If A = [aij], B = [bij] are matrices of the same order, say m × n, then:

A + B = B + A

i.e. A + B = [aij] + [bij]

= [aij + bij]

= [bij + aij] (addition of numbers is commutative)

= ([bij] + [aij])

= B + A

Therefore, the addition of matrices is commutative.

[2] Associative Law: For any three matrices A = [aij], B = [bij], C = [cij] of the same order, say m × n, then:

(A + B) + C = A + (B + C)

This can be shown as:

(A + B) + C = ([aij] + [bij]) + [cij]

= [aij + bij] + [cij]

= [(aij + bij) + cij]

= [aij + (bij + cij)] (addition of numbers is associative)

= [aij] + [(bij + cij)]

= [aij] + ([bij] + [cij])

= A + (B + C)

[3] Existence of additive identity: Let A = [aij] be an m × n matrix and O be an m × n zero matrix, then:

A + O = O + A = A

[4] The existence of additive inverse: Let A = [aij] be any matrix of the order m × n, then we have another matrix as –A = [–aij]m × n such that A + (–A) = (–A) + A = O.

Thus, –A is the additive inverse of A or negative of A. The negative of a matrix is denoted by –A and it can be defined as –A = (–1) A.

Now, by equating the corresponding elements,

2x + 2 = 8

2x = 6

x = 3

Also, 2 + y = 5

y = 3

Therefore, x = 3 and y = 3.

Subtraction of matrices:

The Subtraction of matrices is possible only when the order of the two matrices is the same.

but for multiplication of matrices, we need to check if the number of columns of one matrix is equal to the number of rows of the second matrix.

If there are two matrices, say A = [aij] and B = [bij] of the same order, say m × n, then the subtraction of A and B, i.e., A – B is defined as:

Matrix D = [dij]

A – B = aij – bij

Thus,

dij = aij – bij, (i = 1,2,3,… and j= 1,2,3…)

D = A – B = aij – bij

A – B = A + (-B)

### Subtraction of 2 x 2 Matrices

Suppose A and B are 2 x 2 matrices, such that;

Then, subtraction of matrices A and B, will be given as:

Fact: If A and B are two matrices of the same order, then;

A – B ≠ B – A

Thus, commutative law is not applicable for subtraction of matrices.

### Subtraction of 3 x 3 Matrices

Suppose A and B are 3 x 3 matrices, such that;

Then, subtraction of matrices A and B, will be given as:

## Examples

Q.1: If A  and B are two matrices. Then find subtraction of matrices A and B.

## Matrix Multiplication Definition

Matrix multiplication, also known as matrix product and the multiplication of two matrices, produces a single matrix. It is a type of binary operation.

If A and B are the two matrices, then the product of the two matrices A and B are denoted by:

X = AB

Hence, the product of two matrices is the dot product of the two matrices.

Matrix multiplication by Scalar

Multiplication of an integer with a matrix is simply a scalar multiplication.

We know that a matrix is an array of numbers. It consists of rows and columns. If you multiply a matrix by a scalar value, then it is known as scalar multiplication. Another case is that it is possible to multiply a matrix by another matrix. Let’s have a look at the example given below for the same.

We may define multiplication of a matrix by a scalar mathematically as:

If A = [aij]m × n is a matrix and k is a scalar, then kA is another matrix obtained by multiplying each element of A by the scalar k.

In other words, kA = k [aij]m × n = [k (aij)]m × n, that is, (i, j)th element of kA is kaij for all possible values of i and j.

Example: Multiply the matrix

Matrix multiplication Condition

To perform multiplication of two matrices, we should make sure that the number of columns in the 1st matrix is equal to the rows in the 2nd matrix. Therefore, the resulting matrix product will have a number of rows of the 1st matrix and a number of columns of the 2nd matrix. The order of the resulting matrix is the matrix multiplication order.

Four  Types of multiplication :

i) row by row

ii) row by column

iii) column by column

iv) column by row

Normally we multiply row by column method.

## How to Multiply Matrices?

Let’s learn how to multiply matrices.

Consider matrix A which is a × b matrix and matrix B, which is a b ×c matrix.

Then, matrix C = AB is defined as the A × B matrix.

An element in matrix C, Cxy is defined as Cxy = Ax1By1 +….. + AxbBby =

AxkBky  for x = 1…… a  and y= 1…….c

### Notation

If A is a m×n matrix and B is a p×q matrix, then the matrix product of A and B is represented by:

X = AB

Where X is the resulting matrix of m×q dimension.

Matrix Multiplication Formula

Let’s take an example to understand this formula.

Let’s say A and B are two matrices, such that,

Then Matrix C = AB is denoted by

An element in matrix C where C is the multiplication of Matrix A X B.

C = Cxy = Ax1By1 +….. + AxbBby =(AB)ij=Cij=r=1nairbrj

AxkBky  for x = 1…… a  and y= 1…….c

### Matrix multiplication Rules

Rules and properties for matrix multiplication.

• The product of two matrices A and B is defined if the number of columns of A is equal to the number of rows of B.
• If AB is defined, then BA need not be defined
• If both A and B are square matrices of the same order, then both AB and BA are defined.
• If AB and BA are both defined, it is not necessary that AB = BA.
• If the product of two matrices is a zero matrix, it is not necessary that one of the matrices is a zero matrix.

## 2×2 Matrix Multiplication

Let a simple 2 × 2 matrix multiplication

Let a simple 2 × 2 matrix multiplication

And  another  matrix

Now each of the elements of product matrix AB can be calculated as follows:

• AB11 = 3 × 6 + 7 ×5 = 53
• AB12 = 3 × 2 + 7 × 8 = 62
• AB21 = 4 × 6 + 9 × 5 = 69
• AB22 = 4 × 2 + 9 × 8 = 80

3×3 Matrix Multiplication

The multiplication of two 3 × 3 matrices, let two 3 × 3 matrices A and B.

Each element of the Product matrix AB can be calculated as follows:

• AB11 = 12×5 + 8×6 + 4×7 = 136
• AB12 = 12×19 + 8×15 + 4×8 = 380
• AB13 = 12×3 + 8×9+4×16 = 172
• AB21 = 3×5 + 17×6 + 14×7 = 215
• AB22 = 3×19 + 17×15 + 14×8 = 424
• AB23 = 3×3 + 17×9 + 14×16 = 386
• AB31 = 9×5 + 8×6 + 10×7 = 163
• AB32 = 9×19 + 8×15 + 10×8 = 371
• AB33 = 9×3 + 8×9 + 10×16 = 259

## Properties of Matrix Multiplication

The following are the properties of the matrix multiplication:

### Commutative Property

The matrix multiplication is not commutative.

Assume that, if A and B are the two 2×2 matrices,

AB ≠ BA

In matrix multiplication, the order matters a lot.

For example,

This shows that the matrix AB ≠BA.

Hence, the multiplication of two matrices is not commutative.

### Associative Property

If A, B and C are the three matrices, the associative property of matrix multiplication states that,

(AB) C = A(BC)

Hence, the associative property of matrix multiplication is proved.

### Distributive Property

If A, B and C are the three matrices, the distributive property of matrix multiplication states that,

• (B+C)A = BA +CA
• A(B+C) = AB + AC

### Multiplicative Identity Property

The identity property of matrix multiplication states that,

### Dimension Property

In matrix multiplication, the product of m × n matrix and n×a matrix is the m× a matrix.

For example, matrix A is a 2 × 3 matrix and matrix B is a 3 × 4 matrix, then AB is a 2 × 4 matrices.

### Multiplicative property of Zero

If a matrix is multiplied by a zero matrix, the result matrix is a zero matrix.

## 3. Transpose of a matrices and Properties of transpose of the matrices

### Transpose of a Matrix

If A = [aij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A and is denoted by A′ or (AT ).

In other words, if A = [aij] m × n , then A′ = [aji] n × m .

Example:

Properties of Transpose of a Matrix

## 4. Symmetric and Skew Symmetric Matrices

### Symmetric and Skew Symmetric Matrices

A square matrix A = [aij] is said to be symmetric if the transpose of A is equal to A, that is, [aij] = [aji] for all possible values of i and j.

A square matrix A = [aij] is a skew-symmetric matrix if A′ = – A, that is aji = – aij for all possible values of i and j. Also, if we substitute i = j, we have aii = – aii and thus, 2aii = 0 or aii = 0 for all i’s. Therefore, all the diagonal elements of a skew symmetric matrix are zero.

Properties of Symmetric Matrix

• Addition and difference of two symmetric matrices results in symmetric matrix.
• If A and B are two symmetric matrices and they follow the commutative property,

i.e. AB =BA, then the product of A and B is symmetric.

• If matrix A is symmetric then An is also symmetric, where n is an integer.
• If A is a symmetrix matrix then A-1 is also symmetric.

### Properties of Skew Symmetric Matrix

• When we add two skew-symmetric matrices then the resultant matrix is also skew-symmetric.
• Scalar product of skew-symmetric matrix is also a skew-symmetric matrix.
• The diagonal of skew symmetric matrix consists of zero elements and therefore the sum of elements in the main diagonals is equal to zero.
• When identity matrix is added to skew symmetric matrix then the resultant matrix is invertible.
• The determinant of skew symmetric matrix is non-negative

Theorem :  How do you prove symmetric and skew symmetric matrix?

Example

Properties of Symmetric and Skew-Symmetric Matrices

There are some rules that come from the concept of Symmetric and Skew-Symmetric Matrices,

1. If matrix A is a square matrix then (A + At) is always symmetric.

Prove:

To find if a matrix symmetric or not, first, we have to find the transposed form of the given matrix

So, let’s find the transpose of (A + At)

= (A + At)t

= A+ (At)t

= A+ A   [here, (At)= A]

= (A + At)

So, this is the same as the given matrix, so it is symmetric.

2. If matrix A is a square matrix then (A – At) is always skew-symmetric.

Prove:

To find if a matrix skew-symmetric or not, first, we have to find the transposed form of the given matrix

So, let’s find the transpose of (A – At)

= (A − At)t

= A− (At)t

= A− A  [here, (At)t = A]

= − (A − At

So, this form is the negative of the given matrix, so it is skew-symmetric.

Theorem 1:

Any Square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix. Proof: Let A be a square matrix then, we can write A = 1/2 (A + A′) + 1/2 (A − A′). From the Theorem 1, we know that (A + A′) is a symmetric matrix and (A – A′) is a skew-symmetric matrix.

Theorem 2: Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.

Proof: Let A = [aij]mxm

We know that A + A' : Symmetric matrix

A - A' : Skew Symmetric matrix

½(A + A' ) + ½ (A - A')

= (½A + ½A') + (½A - ½A') Using kA = k(A)

½A + ½A = A.

Theorem 3: Inverse of matrix A is unique.

Proof:

Let B and C be two inverses of A.

i.e., AxB=BxA=I

and  AxC =CxA=1

Now, B=BI = B (AxC)  = ( BxA) X C      = I xC    =     C

So, inverses are unique.

## 5. Invertible Matrices and Inverse of a matrix by elementary operations

Invertible Matrices and Inverse of a matrix by elementary operations

Elementary Operation (Transformation) of a Matrix

There are six operations (transformations) on a matrix, three of which are due to rows, and three are due to columns, known as elementary operations or transformations.

[1] The interchange of any two rows or two columns.

denoted by Ri ↔ Rj and interchange of i th and j th column is denoted by Ci ↔ Cj .

[2] The multiplication of the elements of any row or column by a non zero number.

denoted by Ri → kRi . The corresponding column operation is denoted by Ci → kCi

[3] The addition to the elements of any row or column, the corresponding elements of any other row or column are multiplied by any non zero number.

denoted by Ri → Ri + kRj . The corresponding column operation is denoted by Ci → Ci + kCj .

Elementary transformations are operations done on the rows and columns of matrices to change their shape so that the computations become easier. It is also used to discover the inverse of a matrix, the determinants of a matrix, and to solve a system of linear equations. A square matrix is always an elementary matrix.

Invertible Matrices

Suppose a square matrix A of order m, and if there exists another square matrix B of the same order m, such that AB = BA = I, then B is called the inverse matrix of A, and it is denoted by A-1. Also, matrix A is said to be an invertible matrix here.

Note:-

[1] A rectangular matrix does not possess inverse matrix, since for products BA and AB to be defined and to be equal, it is necessary that matrices A and B should be square matrices of the same order.

[2] If B is the inverse of A, then A is also the inverse of B.

Theorem 4:  If A and B are invertible matrices of the same order,

then (AB)–1 = B–1 A–1

Proof : From the definition of inverse of a matrix,

we have (AB) (AB)–1 = I

or A–1 (AB) (AB)–1 =A–1I (Pre multiplying both sides by A–1)

or (A–1A) B (AB)–1 =A–1 (Since A–1 I = A–1)

or IB (AB) –1  =A–1  or B (AB) –1  =A–1

or B–1  B (AB) –1  =B–1 A–1

or I (AB) –1  =B–1  A–1

Hence (AB) –1  =B–1  A–1

Example 1:

, then find the value of a, b, c, x, y, and z.

Solution:

It is given that, the two matrices are equal. Therefore, the corresponding elements present in matrices should be equal to each other. By comparing the corresponding elements in the matrices, we get:

x+3 = 0. ⇒ x = -3

z +4 = 6  ⇒ z = 6-4

⇒ z = 2

2y-7 = 3y-2     ⇒3y-2y =-7+2

⇒y = -5

a-1 = -3

⇒a = -3+1

⇒a=-2

2c+2 = 0

⇒2c = -2

⇒ c = -1

b-3 = 2b+4

⇒2b-b = -3-4

⇒ b = -7

Therefore, the values of the variables are:

a = -2

b = -7

c = -1

x = -3

y = -5

z = 2

Example 2:

Now, we need to calculate the transpose of AB.

Hence  verified.

## 2. Area of a Triangle

Area of a Triangle

The area of a triangle formed by the vertices - (x1, y1), (x2, y2) and (x3, y3) is given by

½[x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)] sq.units

It can be expressed in the form of a determinant.

Note: We always take 1 in the last column of the determinant.

Expanding along the first column, we get

= x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)

The area of a triangle formed by the vertices - (x1, y1), (x2, y2) and (x3, y3) –

Note:

In calculating the area of a triangle by using this formula, we need to take the absolute value of the above determinant to avoid negative values, if any.

If area is given, use both positive and negative values of the determinant for the calculation.

The area of a triangle formed by three collinear points is equal to zero.

Example: Find out the area of the triangle whose vertices are given by A(0,0) , B (3,1) and C (2,4).

Solution: Using determinants we can find out the area of the triangle obtained by joining these points using the formula

Example : If (k, 2), (2, 4) and (3, 2) are vertices of the triangle of area 4 square units then determine the value of k.

Solution:

Area of triangle  = 4 square units

• (1/2){k [4 – 2] – 2[2 – 3] + 1[4 – 12]} = 4
• k(2) – 2(-1) + 1(-8) = 8
• 2k + 2 – 8 = 8
• 2k – 6 = 8
• 2k = 8 + 6
• 2k = 14      => k = 7

So, the value of k is 7.

• (1/2){k [4 – 2] – 2[2 – 3] + 1[4 – 12]} = -4
• k(2) – 2(-1) + 1(-8) = -8
• 2k + 2 – 8 = -8
• 2k – 6 = -8
• 2k = -8 + 6
• 2k = -2      => k = -1

So, the value of k is -1.

## 3. Minors and Co-factors

Minors and Cofactors:

Minor of an element in a matrix is defined as the determinant obtained by deleting the row and column in which that element lies. e.g. in the determinant

Row and Column Operations :

## 4. Adjoint and Inverse of a Matrix

Adjoint and Inverse of a Matrix

The adjoint of a matrix (also called the adjugate of a matrix) is defined as the transpose of the cofactor matrix of that particular matrix. For a matrix A, the adjoint is denoted as adj (A). On the other hand, the inverse of a matrix A is that matrix which when multiplied by the matrix A give an identity matrix.

Note.: For a square matrix of order 2, given by

Let the determinant of a square matrix A be     |A|

Then the transpose of the matrix of co-factors is called the adjoint of the matrix A and is written as adj( A )

The product of a matrix A and its adjoint is equal to unit matrix multiplied by the determinant A.

Let A be a square matrix, then (Adjoint A). A = A. (Adjoint A) = | A |. I

Note:

Theorem  : If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order.

TheoremThe determinant of the product of matrices is equal to product of their respective determinants, that is, |AB| = |A|| B| , where A and B are square matrices of the same order.

Theorem : If A is a square matrix of order n, then |adj(A)| = |A|n – 1.

Let A and B  be two non singular square matrix of same order n,

• Property 1:  A (adj(A)) = (adj(A)) A = |A|I, where I is the identitiy matrix of order n.
• Property 2: A square matrix A is invertible if and only if A is a non-singular matrix.

Example : If AT = – A then the elements on the diagonal of the matrix are equal to

(a) 1 (b) -1 (c) 0 (d) none of these

Solution:

(c) AT = -A; A is skew-symmetric matrix; diagonal elements of A are zeros.

so option (c) is the answer.

Example : If A and B are two skew-symmetric matrices of order n, then,

(a) AB is a skew-symmetric matrix (b) AB is a symmetric matrix

(c) AB is a symmetric matrix if A and B commute (d)None of these

Solution:

(c) We are given A’ = -A and B’ = -B;

Now, (AB)’ = B’A’ = (-B) (-A) = BA = AB, if A and B commute.

Example : Let A and B be two matrices such that AB’ + BA’ = 0. If A is skew symmetric ,then BA

(a) Symmetric (b) Skew symmetric (c) Invertible (d) None of these

Solution:

(c) we have, (BA)’ = A’B’ = -AB’ [ A is skew symmetric]; = BA’ = B(-A)

= -BA

BA is skew symmetric.

Example : Which of the following statements are false –

(a) If | A | = 0, then | adj A | = 0;

(b) Adjoint of a diagonal matrix of order 3 × 3 is a diagonal matrix;

(c) Product of two upper triangular matrices is an upper triangular matrix;

Solution:

Inverse of a Matrix:

If A and B are two square matrices of the same order, such that AB = BA = I (I = unit matrix),Then B is called the inverse of A,

i.e. B = A–1 and A is the inverse of B.

Condition for a square matrix A to possess an inverse is that the matrix A is non-singular, i.e., | A | ≠ 0. If A is a square matrix and B is its inverse then AB = I.

Taking determinant of both sides | AB | = | I | or | A | | B | = I.

From this relation it is clear that | A | ≠ 0, i.e. the matrix A is non-singular.

We know that,

Properties of Inverse matrix:

Let A and B  be two non singular square matrix of same order n,

• Property 1:  (A-1)-1 = A
• Property 2: (AB)-1=B-1A-1
• Property 3: (A’)-1= (A-1)’
• Property 4: (Am)-1= (A-1)m
• Property 5: |A-1|= |A|-1

Example:

=-28+30+18

=20

Similarly we can also obtain the values of B-1 and A-1 Then by multiplying B-1 and A-1

## 5. Applications of Determinants and Matrices

Applications of Determinants and Matrices:

Consistency of System of Equations

Consistent system A system of equations is said to be consistent if its solution (one

or more) exists.

Inconsistent system A system of equations is said to be inconsistent if its solution

does not exist.

Solution of system of linear equations using inverse of a matrix

Suppose the system of equations is given by:

a1 x + b1 y + c1 z = d1

a2 x + b2 y + c2 z = d2

a3 x + b3 y + c3 z = d3

Now let us say, A, B and X are three matrices, such that;

or  I X = A–1 B

or  X = A–1 B

If A is a non-singular matrix, then X = A-1B.

This matrix equation provides unique solution for the given system of equations as inverse of a matrix is unique. This method of solving system of equations is known as Matrix Method.

Case-2 :

If A is a singular matrix, then determinant of A, |A| = 0.

Now for such a condition, there exist two cases based on (adj A) B.

• If (adj A) B  O, (O being is zero matrix), then the system of equations does not have a solution and hence is called inconsistent.
• If (adj A) B = O, then the system of equations will have either consistent or inconsistent according as the system have either infinitely many solutions or no solution.

Problem :

Find if the given system of equations is consistent or inconsistent.

x+3y = 5 and 2x + 6y = 8

Solution: Given, the system of equations are:

x+3y = 5 and 2x + 6y = 8

As per the matrix Method, we know;

AX = B