If the matrix A = \begin{pmatrix} 3 & 2 \\ 7 & 5 \end{pmatrix}, which matrix represents th… — JAMB UTME

If the matrix $A = \begin{pmatrix} 3 & 2 \\ 7 & 5 \end{pmatrix}$ , which matrix represents the inverse $A^{-1}$ ?

\begin{pmatrix} 5 & -2 \\ -7 & 3 \end{pmatrix}

The correct inverse matrix is computed by evaluating the determinant

\det(A) = 3(5) - 2(7) = 1

and constructing the adjugate matrix by swapping the diagonal elements

3

and

5

while changing the signs of

2

and

7

, resulting in

\begin{pmatrix} 5 & -2 \\ -7 & 3 \end{pmatrix}

Calculate the determinant of matrix

A

\det(A) = (3)(5) - (2)(7) = 15 - 14 = 1

The inverse requires dividing the adjugate matrix by the determinant of

A

Find the adjugate of matrix

A

\text{adj}(A) = \begin{pmatrix} 5 & -2 \\ -7 & 3 \end{pmatrix}

For a 2x2 matrix

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

, swap the main diagonal elements (

a

and

d

) and change the signs of the off-diagonal elements (

b

and

c

Compute

A^{-1} = \frac{1}{\det(A)} \text{adj}(A)

A^{-1} = \frac{1}{1} \begin{pmatrix} 5 & -2 \\ -7 & 3 \end{pmatrix} = \begin{pmatrix} 5 & -2 \\ -7 & 3 \end{pmatrix}

Multiply the adjugate matrix by the reciprocal of the determinant.

2x2 Matrix Inversion

Estimated Time:1m 0s

Question