Compute the following:
Question:

Compute the following:

(i) $\left[\begin{array}{rr}a & b \\ -b & a\end{array}\right]+\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]$

(ii) $\left[\begin{array}{ll}a^{2}+b^{2} & b^{2}+c^{2} \\ a^{2}+c^{2} & a^{2}+b^{2}\end{array}\right]+\left[\begin{array}{cc}2 a b & 2 b c \\ -2 a c & -2 a b\end{array}\right]$

(iii) $\left[\begin{array}{lll}-1 & 4 & -6 \\ 8 & 5 & 16 \\ 2 & 8 & 5\end{array}\right]+\left[\begin{array}{lll}12 & 7 & 6 \\ 8 & 0 & 5 \\ 3 & 2 & 4\end{array}\right]$

(v) $\left[\begin{array}{cc}\cos ^{2} x & \sin ^{2} x \\ \sin ^{2} x & \cos ^{2} x\end{array}\right]+\left[\begin{array}{cc}\sin ^{2} x & \cos ^{2} x \\ \cos ^{2} x & \sin ^{2} x\end{array}\right]$

Solution:

(i)

$\left[\begin{array}{rr}a & b \\ -b & a\end{array}\right]+\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]=\left[\begin{array}{ll}a+a & b+b \\ -b+b & a+a\end{array}\right]=\left[\begin{array}{ll}2 a & 2 b \\ 0 & 2 a\end{array}\right]$

(ii)

$\left[\begin{array}{ll}a^{2}+b^{2} & b^{2}+c^{2} \\ a^{2}+c^{2} & a^{2}+b^{2}\end{array}\right]+\left[\begin{array}{cc}2 a b & 2 b c \\ -2 a c & -2 a b\end{array}\right]$

$=\left[\begin{array}{ll}a^{2}+b^{2}+2 a b & b^{2}+c^{2}+2 b c \\ a^{2}+c^{2}-2 a c & a^{2}+b^{2}-2 a b\end{array}\right]$

$=\left[\begin{array}{ll}(a+b)^{2} & (b+c)^{2} \\ (a-c)^{2} & (a-b)^{2}\end{array}\right]$

(iii) $\left[\begin{array}{lll}-1 & 4 & -6 \\ 8 & 5 & 16 \\ 2 & 8 & 5\end{array}\right]+\left[\begin{array}{lll}12 & 7 & 6 \\ 8 & 0 & 5 \\ 3 & 2 & 4\end{array}\right]$

$=\left[\begin{array}{ccc}-1+12 & 4+7 & -6+6 \\ 8+8 & 5+0 & 16+5 \\ 2+3 & 8+2 & 5+4\end{array}\right]$

$=\left[\begin{array}{lll}11 & 11 & 0 \\ 16 & 5 & 21 \\ 5 & 10 & 9\end{array}\right]$

(iv) $\left[\begin{array}{cc}\cos ^{2} x & \sin ^{2} x \\ \sin ^{2} x & \cos ^{2} x\end{array}\right]+\left[\begin{array}{cc}\sin ^{2} x & \cos ^{2} x \\ \cos ^{2} x & \sin ^{2} x\end{array}\right]$

$=\left[\begin{array}{cc}\cos ^{2} x+\sin ^{2} x & \sin ^{2} x+\cos ^{2} x \\ \sin ^{2} x+\cos ^{2} x & \cos ^{2} x+\sin ^{2} x\end{array}\right]$

$=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right] \quad\left(\because \sin ^{2} x+\cos ^{2} x=1\right)$

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