Inverse Matrix Formula

Inverse of a matrix is an important operation in the case of a square matrix. It is applicable only for a square matrix. To calculate the inverse, one has to find out the determinant and adjoint of that given matrix. Adjoint is given by the transpose of cofactor of the particular matrix. The formula to find out the inverse of a matrix is given as,
Inverse Matrix Formula

Inverse Matrix Problems

Let us discuss how to find out inverse of a matrix.

Solved Examples

Question 1: Find the inverse of \(\begin{bmatrix} 5& 6& \\ -1& 2 & \end{bmatrix}\)?

Solution:

Let A = \(\begin{bmatrix} 5& 6& \\ -1& 2 & \end{bmatrix}\) be the given matrix.$A^{-1}$ = $\frac{adj(A)}{| A |}$

To find out the adj(A), first we have to find out cofactor(A).So, cofactor(A) = \(\begin{bmatrix} 2& -6& \\ 1& 5 & \end{bmatrix}\)adj(A) = $[cofactor(A)]^{T}$adj(A) = $[cofactor(A)]^{T}$ = \(\begin{bmatrix} 2& -6 \\ 1& 5 \end{bmatrix}^{T}\)

adj(A) = \(\begin{bmatrix} 2& 1 \\ -6& 5 \end{bmatrix}\)

Then, $| A |$ = (5$\times$2)-(-1$\times$6) = 10 + 6 = 16

$A^{-1}$ = $\frac{adj(A)}{| A |}$ =\(\frac{\begin{bmatrix} 2&1\\ -6& 5 \end{bmatrix}}{16}\)

 

Question 2: Find out the inverse of \(\begin{bmatrix} 1 &-1 &2 \\ 4&0 &6 \\ 0&1 &-1 \end{bmatrix}\)?

Solution:

Let A = \(\begin{bmatrix} 1 &-1 &2 \\ 4&0 &6 \\ 0&1 &-1 \end{bmatrix}\)  be the given matrix.
\(A^{-1} = \frac{adj(A)}{| A |}\)
To find out the adj(A), first we have to find out cofactor(A).
a11 = -6, a12 = 4, a13 = 4
a21 = 1, a22 = -1, a23 = -1
a13 = -6, a32 = 2, a33 = 4So, cofactor(A) = \(\begin{bmatrix} -6&4 &4\\ 1&-1 &-1\\ -6&2 &4 \end{bmatrix}\)

adj(A) = $[cofactor(A)]^{T}$

adj(A) = \([cofactor(A)]^{T} = \begin{bmatrix} -6&4 &4 \\ 1&-1 &-1 \\ -6&2 &4 \end{bmatrix}^{T}\)

adj(A) = \(\begin{bmatrix} -6&1 &-6 \\ 4&-1 &2 \\ 4&-1 &4 \end{bmatrix}\)

Then, | A | = 1(0-6)+1(-4-0)+2(4-0) = -6-4+8 = -2

\(A^{-1} = \frac{adj(A)}{| A |} = \frac{\begin{bmatrix} -6&1 &-6\\ 4&-1 &2\\ 4&-1 &4 & \end{bmatrix}}{-2}\)

Practise This Question

In the following four periods
(i) Time of revolution of a satellite just above the earth’s surface (Tst)
(ii) Period of oscillation of mass inside the tunnel bored along the diameter of the earth (Tma)
(iii) Period of simple pendulum having a length equal to the earth’s radius in a uniform field of 9.8 N/k (Tsp)
(iv) Period of an infinite length simple pendulum in the earth’s real gravitational field (Tis)