Chapter 12.1, Problem 4STP

For Exercises 1-5, choose the correct letter.

What is the value of x in the equation $2\left[\begin{array}{l}-69\\ -31\end{array}\right]-\left[\begin{array}{l}-4-6\\ -53\end{array}\right]=\left[\begin{array}{l}-8x\\ -1-1\end{array}\right]$?

F. -24

G. -12

H. 12

I. 24

To determine

Find the value of x.

Answer to Problem 4STP

I

Explanation of Solution

Given:

  $2\left[\begin{array}{l}\text{-6 9}\\ \text{-3 1}\end{array}\right]-\left[\begin{array}{l}\text{-4 -6}\\ \text{-5 3}\end{array}\right]=\left[\begin{array}{l}\text{-8 x}\\ \text{-1 -1}\end{array}\right]$

Formula Used:

  $x\left[\begin{array}{l}\text{a b}\\ \text{d e }\end{array}\right]=\left[\begin{array}{l}\text{xa xb}\\ \text{xd xe }\end{array}\right]$

  $\left[\begin{array}{l}\text{a b}\\ \text{c d}\end{array}\right]+\left[\begin{array}{l}\text{e f}\\ \text{g h}\end{array}\right]=\left[\begin{array}{l}\text{a+e b+f}\\ \text{c+g d+h}\end{array}\right]$

Calculation:

  $\begin{array}{l}2\left[\begin{array}{l}\text{-6 9}\\ \text{-3 1}\end{array}\right]-\left[\begin{array}{l}\text{-4 -6}\\ \text{-5 3}\end{array}\right]=\left[\begin{array}{l}\text{-8 x}\\ \text{-1 -1}\end{array}\right]\\ \Rightarrow \left[\begin{array}{l}\text{-12 18}\\ \text{ -6 2}\end{array}\right]+\left[\begin{array}{l}\text{4 6}\\ \text{5 -3}\end{array}\right]=\left[\begin{array}{l}\text{-8 x}\\ \text{-1 -1}\end{array}\right]\mathrm{..................}\left\{x\left[\begin{array}{l}\text{a b}\\ \text{d e }\end{array}\right]=\left[\begin{array}{l}\text{xa xb}\\ \text{xd xe }\end{array}\right]\right\}\\ \Rightarrow \left[\begin{array}{l}\text{-12+4 18+6}\\ \text{ -6+5 2-3}\end{array}\right]=\left[\begin{array}{l}\text{-8 x}\\ \text{-1 -1}\end{array}\right]\mathrm{...........................}\left\{\left[\begin{array}{l}\text{a b}\\ \text{c d}\end{array}\right]+\left[\begin{array}{l}\text{e f}\\ \text{g h}\end{array}\right]=\left[\begin{array}{l}\text{a+e b+f}\\ \text{c+g d+h}\end{array}\right]\right\}\\ \Rightarrow \left[\begin{array}{l}\text{-8 24}\\ \text{-1 -1}\end{array}\right]=\left[\begin{array}{l}\text{-8 x}\\ \text{-1 -1}\end{array}\right]\\ \Rightarrow x=\mathrm{24.................................................................}[\text{two matrices are equal if the corresponding values are equal]}\end{array}$

So, the correct option is I.