.0. If A=(} 3&1 4&2 ) evaluate: (ii) (A+B)^2

To evaluate \((A+B)^2\), we first need to find the sum of matrices \(A\) and \(B\), and then square the resulting matrix.<br /><br />Given:<br />\[ A = \begin{pmatrix} 3 & 1 \\ 4 & 2 \end{pmatrix} \]<br />\[ B = \begin{pmatrix} 5 & 1 \\ 3 & 4 \end{pmatrix} \]<br /><br />First, compute \(A + B\):<br />\[ A + B = \begin{pmatrix} 3 & 1 \\ 4 & 2 \end{pmatrix} + \begin{pmatrix} 5 & 1 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 3+5 & 1+1 \\ 4+3 & 2+4 \end{pmatrix} = \begin{pmatrix} 8 & 2 \\ 7 & 6 \end{pmatrix} \]<br /><br />Next, compute \((A + B)^2\):<br />\[ (A + B)^2 = \begin{pmatrix} 8 & 2 \\ 7 & 6 \end{pmatrix} \times \begin{pmatrix} 8 & 2 \\ 7 & 6 \end{pmatrix} \]<br /><br />To multiply the matrices, use the formula for matrix multiplication:<br />\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \times \begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{pmatrix} \]<br /><br />So,<br />\[ (A + B)^2 = \begin{pmatrix} 8 \cdot 8 + 2 \cdot 7 & 8 \cdot 2 + 2 \cdot 6 \\ 7 \cdot 8 + 6 \cdot 7 & 7 \cdot 2 + 6 \cdot 6 \end{pmatrix} = \begin{pmatrix} 64 + 14 & 16 + 12 \\ 56 + 42 & 14 + 36 \end{pmatrix} = \begin{pmatrix} 78 & 28 \\ 98 & 50 \end{pmatrix} \]<br /><br />Therefore, the final result is:<br />\[ (A + B)^2 = \begin{pmatrix} 78 & 28 \\ 98 & 50 \end{pmatrix} \]