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Matrices — Mathematics STD 10 — Question

ICSE BoardEnglish MediumSTD 10MathematicsMatrices4 Marks
Question
In the given case below, Find $:$
$a)$ The order of matrix $M$
$b)$ The matrix $M \left[\begin{array}{ll}1 & 4 \\ 2 & 1\end{array}\right] \times M=\left[\begin{array}{c}13 \\ 5\end{array}\right]$
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Answer

Let the order of matrix $M$ be $a \times b$
$\left[\begin{array}{ll}1 & 4 \\ 2 & 1\end{array}\right]_{2 \times 2} \times M_{a \times b}=\left[\begin{array}{c}13 \\ 5\end{array}\right]_{2 \times 1}$
Clearly the order of matrix $M$ is $2 \times 1$
Let $M=\left[\begin{array}{l}a \\ b\end{array}\right]$
$\left[\begin{array}{ll}1 & 4 \\ 2 & 1\end{array}\right] \times M=\left[\begin{array}{c}13 \\ 5\end{array}\right]$
$\left[\begin{array}{ll}1 & 4 \\ 2 & 1\end{array}\right] \times\left[\begin{array}{l}a \\ b\end{array}\right]=\left[\begin{array}{c}13 \\ 5\end{array}\right]$
$\left[\begin{array}{l}a+4 b \\ 2 a+b\end{array}\right]=\left[\begin{array}{c}13 \\ 5\end{array}\right]$
Comparing the corresponding elements we get
$a + 4b = 13 .....(1)$
$2a + b = 5....(2)$
Multiplying $(2)$ by $4$ we get
$8a + 4b = 20 ....(3)$
Substracting $(1)$ from $(3)$ we get
$a=7=>a=1$
From $(2)$ we get
$b = 5 - 2a = 5- 2 = 3$
$\therefore M=\left[\begin{array}{l}a \\ b\end{array}\right]=\left[\begin{array}{l}1 \\ 3\end{array}\right]$

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