If T={pmatrix} -2 & -5 \\ 3 & 8 {pmatrix}, find T^{-1}, the inverse of T .

Correct Answer is : (−8−532){pmatrix} -8 & -5 \\ 3 & 2 {pmatrix}(−83−52)

{pmatrix} -8 & -5 \\ 3 & 2 {pmatrix}

{pmatrix} -8 & -5 \\ 3 & -2 {pmatrix}

{pmatrix} -8 & -5 \\ -3 & 2 {pmatrix}

{pmatrix} -8 & -5 \\ -3 & -2 {pmatrix}

Correct Answer is : (−8−532){pmatrix} -8 & -5 \\ 3 & 2 {pmatrix}(−83−52)

Test Index

WAEC Further Mathematics 2014 Paper

Question : 14

Total: 40

T=\begin{pmatrix} -2 & -5 \\ 3 & 8 \end{pmatrix}

T^{-1}

Solution:

Concept:

For a

2\times 2

matrix

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

, the inverse is given by

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

Explanation:

Here,

T=\begin{pmatrix} -2 & -5 \\ 3 & 8 \end{pmatrix}

, so

a=-2

b=-5

c=3

, and

d=8

First, calculate the determinant:

ad-bc=(-2)(8)-(-5)(3)

This gives

-16+15=-1

Since the determinant is not zero, the inverse exists.

The adjugate matrix is

\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}=\begin{pmatrix} 8 & 5 \\ -3 & -2 \end{pmatrix}

Now multiply by

\frac{1}{ad-bc}=\frac{1}{-1}=-1

Thus,

T^{-1}=(-1)\begin{pmatrix} 8 & 5 \\ -3 & -2 \end{pmatrix}

This simplifies to

T^{-1}=\begin{pmatrix} -8 & -5 \\ 3 & 2 \end{pmatrix}

Answer:

T^{-1}=\begin{pmatrix} -8 & -5 \\ 3 & 2 \end{pmatrix}

, which is option A.

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