Matrix Formula: Explanation, Formulas & Examples

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Matrices are used for solving linear equations. It provides a method for arranging numbers in rows and columns. Various mathematical operations like addition, subtraction, and multiplication can be carried out on these matrices. Matrix formulas are used for calculating transpose, inverse, zero, etc.

Read more: Types of matrices

Table of Content

Transpose Of A Matrix
Properties of a Transpose
Inverse Of A Matrix
Matrix Addition And Subtraction
Multiplication By A Scalar
Determinant Of A Matrix
Things to Remember
Sample Questions

KeyTerms: Matrices, Matrix, Transpose, Inverse matrix, Matrix operations, Multiplication, Linear equations, Addition, Subtraction, Numbers

Transpose of a Matrix

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If we consider a m x n matrix A=[a_ij] , then the transpose of that matrix can be obtained by interchanging the rows and columns of A, called the transpose of A and is denoted by A’ or A^T. Thus, if A=[a_ij]_mxn, then A’=[a_ij]_mxn. thus, one could say that in the transposition of a matrix, the rows become the columns and vice versa. It is also called the adjoint of a matrix.

The video below explains this:

Matrices Detailed Video Explanation:

Properties of a Transpose

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The following properties have been demonstrated for the transpose of a matrix:

(A’)’ = A
(kA)’ = kA
(A+B)’ =A’ + B’
(AB)’ =B’A’

Inverse of a Matrix

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Inverse can only be applied for square matrices. This requires the calculation of the adjoint and determinant of the matrix in question. If we consider a square matrix, A of the order m, another matrix, B can be called the inverse of an if AB = BA = 1. This can be depicted by A^-1. In these cases, A is said to be invertible. Also, like B would be the inverse of A, A would also be the inverse of B.

Rectangular matrices cannot possess an inverse matrix. This is because the product AB and BA could be defined and equal only for square matrices of the same order.

Two theories are discussed in this regard

Inverse of a matrix, if it exists, is unique (Uniqueness of inverse theorem)
If A and B are invertible matrices of the same order, then (AB)^-1 = B^-1 A^-1

Inverse of a 2 x 2 matrix can be given by the formula

M^{– 1} = 1 / \|M\| x adj(M)

Where adj(M) is the adjoint or transpose of the matrix

Matrix Addition and Subtraction

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To carry out operations like addition and subtraction on matrices, they should have exactly the same dimensions, i.e., their size should be the same. Thus, it would be possible to add a matrix 3 x 3 to another 3 x 3 matrix. However, it would be impossible to add it to a 3 x 2 matrix.

Addition and subtraction of matrices can be carried out like regular addition. The number of each position is added or subtracted from the corresponding numbers of the other matrix. The resulting sum or difference would be put up as a new matrix with the corresponding positions being replaced by the sums or differences. The resulting matrix would be of a similar order as the initial matrices that were added.

The addition of matrices follows the following properties:

Commutative law
Associative law
Existence of additive identity
Existence of additive inverse

Read more:

Relevant Concepts
Scalars and Vectors	Matrix addition	Matrix multiplication
Column matrix	Inverse function formula	Invertible matrices

Multiplication of Matrix by a Scalar

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A matrix can be multiplied by a scalar by multiplying each component of the matrix by that particular scalar quantity. Thus, if a matrix A = [a_ij]_mxn is to be multiplied by a scalar k , another matrix kA is obtained where each element of A is multiplied by k.

Determinant of a matrix

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The determinant of a matrix can be given by the formula

\|M\| = m₁₁m₂₂ – m₁₂m₂₁

Things to Remember

Transpose of a matrix is obtained by interchanging the rows and columns
The inverse of a matrix can be given by the formula M^{– 1} = 1 / |M| x adj(M)
Addition of a matrix can be carried out by adding or subtracting the corresponding elements of two matrices of the same order
A matrix can be multiplied by a scalar by multiplying each element of the matrix by that scalar.

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Sample Questions

Ques 1. If [9 -1 4 -2 1 3] = A + [1 2 -1 0 4 9] , then find the matrix A. (Delhi 2013) (2 marks)

Ans. The given matrix can be written as

A= [9 -1 4 -2 1 3] – [1 2 -1 0 4 9]

A= [9-1 -1-2 4 + 1 -2-0 1-4 3-9]

= [8 -3 5 -2 -3 -6]

Ques 2. From the following matrix equation, find the value of x. (2 marks)
[x+y 4 -5 3y] = [3 4 -5 6] (Foreign 2010)

Ans. From the equation, 3y=6

Therefore, y = 6/3 = 2

Also, x + y=3

Or x = 3 -y

Or x = 3 - 2 = 1

Ques 3. Find the values of x y, z and t if
[2x z y t] + [3 1 -1 0 2] = [3 3 5 4 6] (3 marks)

Ans. Here

[2x z y t] + [3 1 -1 0 2] =[3 3 5 4 6]

[2x 2z 2y 2t] +[3 -3 0 6] =[9 15 12 18]

[2xq3 2z-3 2y 2t+6] =[9 15 12 18]

Therefore,

2x + 3 = 9

Or 2x =9 – 3 = 6

Or x = 6/2 = 3

Also, 2z -3 = 15

Or 2z =15 +3 = 18

Or z = 18/2 = 9

Also, 2y = 12

Or y =12/2 = 6

2t + 6 = 18

Or 2t = 18 – 6 = 12

Or t =12 / 2 = 6

Thus, x = 3, y= 6, z = 9 and t = 6

Ques 4. For the matrix A= [2 3 5 7] , find (A+ A') and show that it is a symmetric matrix. (A.I.C.B.S.E 2019) (3 Marks)

Ans. A= [2 3 5 7]

A'=[2 5 3 7]

A+A'=[2 3 5 7] +[2 5 3 7]

=[2+2 3+5] [5+3 7+7]

=[4 8 8 14]

Also, (A +A')’ =[4 8 8 14] = (A+ A')

Hence, ( A+ A') is symmetric.

Ques 5. If A = [5 2 0 9] and B = [3 6 0 -1] , find 3A-2B. (2 marks)

Ans. 3A – 2B = 3[5 2 0 9] – 2[3 6 0 -1]

=[15 6 0 27] - [6 12 0 -2]

=[15-6 6-12 0 27+2]

=[9 -6 0 29]

Ques 6. If [x-y z 2x-y w] =[-1 4 0 5] , then find the value of x+ y. (All India 2014) (3 Marks)

Ans. Here, [x-y z 2x-y w] =[-1 4 0 5]

Thus, x – y = -1

And 2x – y = 0

Solving the equations, we get = 1 and y = 2

Thus, x + y = 1 + 2 = 3

Ques 7. If [x-y 4 z+6 x+y] =[8 w 0 6] , then write the value of (x + y + z). (Delhi 2014 C) (4 Marks)

Ans. Here, [x-y 4 z+6 x+y] =[8 w 0 6] ,

Thus, x – y = 8

And, x + y = 6

On solving the equations, we get x = 7 and y = -1

Also, z + 6 = 0

Or, z = -6

Thus, x + y + z = 7 – 1 – 6 = 0

Ques 8. If 2[1 3 0 x] + [y 0 1 2] = [5 6 1 8] , then find the value of x+ y. (Delhi 2013 C, All India 2012) (3 Marks)

Ans. Here, 2 [1 3 0 x] + [y 0 1 2] = [5 6 1 8]

Thus, [2 6 0 2x] + [y 0 1 2] = [5 6 1 8]

Or [2+y 6 1 2x+2] = [5 6 1 8]

Thus, 2 + y = 5

Or y = 5 – 2 = 3

Also, 2x + 2 =8

Or x + 1 = 4

Or x = 4 – 1 = 3

Thus, x + y = 3 + 3 = 6

Ques 9. Find the value of y – x from following equation
2 [x 5 7 y-3] + [3 -4 1 2] = [7 6 15 14] (All India 2012) (5 marks)

Ans. From the equation,

2 [x 5 7 y-3] + [3 -4 1 2] = [7 6 15 14]

Or [2x 10 14 2y-6] + [3 -4 1 2] = [7 6 15 14]

[2x+3 10-4 14]+[1 2y-6+2] = [7 6 15 14]

Thus,

2x + 3 = 7

Or 2x = 7 – 3 = 4

x = 4 / 2 = 2

Also 2y – 6 + 2 = 14

2y = 14 + 4 = 18

y = 18 / 2 = 9

This y – x = 9 – 2 = 7

Ques 10. If [x² 3] + [y-1 1] =[10 5] , then write the value of x (Foreign 2012) (3 marks)

Ans. Given, [x² 3] + [y-1 1] =[10 5]

[2x 3x] + [-y y] =[10 5]

Thus, 2x – y = 10

And 3x + y = 5

Solving the equations,

We get x = 3

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Matrix Formula: Explanation, Formulas & Examples

Transpose of a Matrix

Matrices Detailed Video Explanation:

Properties of a Transpose

Inverse of a Matrix

Matrix Addition and Subtraction

Multiplication of Matrix by a Scalar

Determinant of a matrix

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

2.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

3.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

4.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

5.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

6.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

Similar Mathematics Concepts

Comments

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Matrix Formula: Explanation, Formulas & Examples

Transpose of a Matrix

Matrices Detailed Video Explanation:

Properties of a Transpose

Inverse of a Matrix

Matrix Addition and Subtraction

Multiplication of Matrix by a Scalar

Determinant of a matrix

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

2. Find: If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

3. Find: The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

4. Find: If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

5.Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

6.If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

2.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

3.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

4.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

5.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

6.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).