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Let = $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ p = \...

Let = $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ p = $\begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}$ Q = $\begin{pmatrix} u & 4+u \\ -2v & v \end{pmatrix}$ be 2 x 2 matrices such that PQ = 1. Find (u, v)

A.

(- $\frac{5}{2}$ - 1)

B.

(- $\frac{5}{2}$ - $\frac{3}{2}$ )

C.

(- $\frac{5}{6}$ - 1)

D.

( $\frac{5}{2}$ - $\frac{3}{2}$ )

View answer & explanation

Correct answer is A

PQ = $\begin{pmatrix} 2 & 3 \\ 4 & 5 \end{pmatrix}$ $\begin{pmatrix} u & 4+u \\ -2v & v \end{pmatrix}$

= $\begin{pmatrix} (2u-6v & 2(4+u) +3v)\\ 4u-10v & 4(4+u)+5v \end{pmatrix}$

= $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

2u - 6v = 1.....(i)

4u - 10v = 0.......(ii)

2(4 + u) + 3v = 0......(iii)

4(4 + u) + 5v = 1......(iv)

2u - 6v = 1 .....(i) x 2

4u - 10v = 0......(ii) x 1

$\frac{\text{4u - 12v = 0}}{\text{-4u - 10v = 0}}$

-2v = 2 = v = -1

2u - 6(-1) = 1 = 2u = 5

u = - $\frac{5}{2}$

∴ (U, V) = (- $\frac{5}{2}$ - 1)

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