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Algebra of Matrices — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices5 Marks
Question
Find the matrix A such that
$\text{A}=\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\2&4&6\end{bmatrix}$
✓

Answer

It is given that:
$\text{A}=\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\2&4&6\end{bmatrix}$
The matrix given on the R.H.S. of the equation is a 2 × 3 matrix and the one given on the L.H.S. of the equation is a 2 × 3 matrix. Therefore, X has to be a 2 × 2 matrix.
$ \text{X}=\begin{bmatrix}\text{a}&\text{c}\\\text{b}&\text{d}\end{bmatrix}$
Therefore, we have:
$ \begin{bmatrix}\text{a}&\text{c}\\\text{b}&\text{d}\end{bmatrix}\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\2&4&6\end{bmatrix}$
$\begin{bmatrix}\text{a}+4\text{c}&2\text{a}+5\text{c}&3\text{a}+6\text{c}\\\text{b}+4\text{d}&2\text{b}+5\text{d}&3\text{b}+6\text{d}\end{bmatrix}=\begin{bmatrix}-7&-8&-9\\2&4&6\end{bmatrix}$
Equating the corresponding elements of the two matrices, we have:
a + 4c = -7, 2a + 5c = -8, 3a + 6c = -9
b + 4d = 2, 2b + 5d = 4, 3b + 6d = 6
Now, a + 4c = -7 ⇒ a = -7 - 4c
$\therefore$ 2a + 5c = -8 ⇒ -14 - 8c + 5c = -8
⇒ -3c = 6
⇒ c = -2
$\therefore$ a = -7 - 4(-2) = -7 + 8 = 1
Now, b + 4d = 2 ⇒ b = 2 - 4d
$\therefore$ 2b + 5d = 4 ⇒ 4 - 8d + 5d = 4
⇒ -3d = 0
⇒ d = 0
$\therefore$ b = 2 - 4(0) = 2
Thus, a = 1, b = 2, c = -2, d = 0
Hence, the required matrix X is $\begin{bmatrix}1&-2\\2&0\end{bmatrix}.$

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