Chapter 5, Problem 13STP

To determine

To calculate: The value of $a$ from the matrix equation,

  $\left[\begin{array}{cc}4& 3\\ 2& 2\end{array}\right]\cdot \left[\begin{array}{c}a\\ b\end{array}\right]=\left[\begin{array}{c}21\\ 9\end{array}\right]$

Answer to Problem 13STP

The value of $a$ is 7.5

Explanation of Solution

Given information:

The matrix equation is given as:

  $\left[\begin{array}{cc}4& 3\\ 2& 2\end{array}\right]\cdot \left[\begin{array}{c}a\\ b\end{array}\right]=\left[\begin{array}{c}21\\ 9\end{array}\right]$

Formula used:

For the matrix of order $2\times 2$ ,

  $A=\left[\begin{array}{ll}p\hfill & q\hfill \\ r\hfill & s\hfill \end{array}\right]$

The determinant is given as $\left|A\right|=ps-qr$ .

The inverse of matrix $A=\left[\begin{array}{ll}p\hfill & q\hfill \\ r\hfill & s\hfill \end{array}\right]$ is $A^{-1}=\frac{1}{\left|A\right|}\left[\begin{array}{cc}s& -q\\ -r& p\end{array}\right]$ where $\left|A\right|=ps-qr\ne 0$

Calculation:

Consider the matrix equation,

  $\left[\begin{array}{cc}4& 3\\ 2& 2\end{array}\right]\cdot \left[\begin{array}{c}a\\ b\end{array}\right]=\left[\begin{array}{c}21\\ 9\end{array}\right]$

Step 1: Solve the inverse of the coefficient matrix as,

  $A^{-1}=\frac{1}{2}\left[\begin{array}{cc}2& -3\\ -2& 4\end{array}\right]$

Step 2: Multiply each side of the matrix equation by the inverse matrix,

  $\frac{1}{2}\left[\begin{array}{cc}2& -3\\ -2& 4\end{array}\right]$

  $\left[\begin{array}{cc}4& 3\\ 2& 2\end{array}\right]\cdot \left[\begin{array}{c}a\\ b\end{array}\right]=\frac{1}{2}\left[\begin{array}{cc}2& -3\\ -2& 4\end{array}\right]\left[\begin{array}{c}21\\ 9\end{array}\right]$

Therefore, solve further

  $\begin{array}{l}\frac{1}{2}\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]\left[\begin{array}{c}a\\ b\end{array}\right]=\frac{1}{2}\left[\begin{array}{c}15\\ -6\end{array}\right]\\ \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}a\\ b\end{array}\right]=\frac{1}{2}\left[\begin{array}{c}15\\ -6\end{array}\right]\end{array}$

Further,

  $\left[\begin{array}{c}a\\ b\end{array}\right]=\frac{1}{2}\left[\begin{array}{c}15\\ -6\end{array}\right]$

Thus, the value of $a$ is 7.5