Find the inverse {pmatrix} 5 & 3 \\ 6 & 4 {pmatrix}

Correct Answer is : ∣2−32−352∣{vmatrix} 2 & -3/2 \\ -3 & 5/2 {vmatrix}2−3−2325

{vmatrix} 2 & -3/2 \\ -3 & -5/2 {vmatrix}

{vmatrix} 2 & -3/2 \\ -3 & 5/2 {vmatrix}

{vmatrix} 2 & 3/2 \\ -3 & 5/2 {vmatrix}

Correct Answer is : ∣2−32−352∣{vmatrix} 2 & -3/2 \\ -3 & 5/2 {vmatrix}2−3−2325

Test Index

JAMB Mathematics 2013 Paper

Question : 19

Total: 47

Find the inverse

\begin{pmatrix} 5 & 3 \\ 6 & 4 \end{pmatrix}

Solution:

Concept:

The inverse of a

2 \times 2

matrix

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

is given by

\frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

, provided the determinant

ad-bc \neq 0

Explanation:

Let

A = \begin{pmatrix} 5 & 3 \\ 6 & 4 \end{pmatrix}

First, find the determinant:

|A| = (5)(4) - (3)(6) = 20 - 18 = 2

Since

|A| \neq 0

, the matrix is invertible.

Swap the diagonal entries and change the signs of the off-diagonal entries:

\begin{pmatrix} 4 & -3 \\ -6 & 5 \end{pmatrix}

Multiply by

\frac{1}{|A|}

, which is

\frac{1}{2}

A^{-1} = \frac{1}{2}\begin{pmatrix} 4 & -3 \\ -6 & 5 \end{pmatrix}

Multiply each entry by

\frac{1}{2}

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

Answer:

The inverse matrix is

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

, which matches option B.

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