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Determinants and Matrices — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsDeterminants and Matrices3 Marks
Question
If $A=\left[\begin{array}{cc}2 & -3 \\ 5 & -4 \\ -6 & 1\end{array}\right], B=\left[\begin{array}{cc}2 & 1 \\ 4 & -1 \\ -3 & 3\end{array}\right]$ and $C=\left[\begin{array}{cc}1 & 2 \\ -1 & 4 \\ -2 & 3\end{array}\right]$ then show that

i. $(A+B)^{\top}=A T+B^{\top}$

✓

Answer

$A+B=\left[\begin{array}{cc}2 & -3 \\ 5 & -4 \\ -6 & 1\end{array}\right]+\left[\begin{array}{cc}2 & 1 \\ 4 & -1 \\ -3 & 3\end{array}\right]$

$=\left[\begin{array}{cc}2+2 & -3+1 \\ 5+4 & -4-1 \\ -6-3 & 1+3\end{array}\right]=\left[\begin{array}{cc}4 & -2 \\ 9 & -5 \\ -9 & 4\end{array}\right]$

$\therefore \quad(A+B)^T=\left[\begin{array}{ccc}4 & 9 & -9 \\ -2 & -5 & 4\end{array}\right]$

$\ldots$ (i)

Now, $\mathbf{A}^{\mathrm{T}}=\left[\begin{array}{ccc}2 & 5 & -6 \\ -3 & -4 & 1\end{array}\right]$ and $\mathbf{B}^{\mathrm{T}}=\left[\begin{array}{ccc}2 & 4 & -3 \\ 1 & -1 & 3\end{array}\right]$

$\therefore \quad \mathrm{A}^{\mathrm{T}}+\mathbf{B}^{\mathrm{T}}=\left[\begin{array}{ccc}2 & 5 & -6 \\ -3 & -4 & 1\end{array}\right]+\left[\begin{array}{ccc}2 & 4 & -3 \\ 1 & -1 & 3\end{array}\right]$

$=\left[\begin{array}{ccc}2+2 & 5+4 & -6-3 \\ -3+1 & -4-1 & 1+3\end{array}\right]$

$=\left[\begin{array}{ccc}4 & 9 & -9 \\ -2 & -5 & 4\end{array}\right]$

...(ii)

From (i) and (ii), we get

$(A+B)^{\top}=A T+B^{\top}$

[Note: The question has been modified.]

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