If \( \mathrm{M}=\left[\begin{array}{cccc}4 & 1 & -3 & 1 \\ 2 & 4 & 3 & -2 \\ 1 & 0 & 1 & 10\end{array}\right] \quad \mathrm{N}=\left[\begin{array}{cccc}2 & -1 & 0 & 5 \\ 4 & 4 & -2 & 0 \\ 1 & 3 & 1 & 5\end{array}\right] \) Find \( 1,(M+N)^{\mathrm{T}} \) \( 2,[M N]^{\mathrm{T}} \)

Let’s tackle both parts of the problem one by one! 1. First, we need to compute \( M + N \). The addition of matrices is performed element-wise, so we add the corresponding elements in \( M \) and \( N \): \[ M + N = \left[\begin{array}{cccc} 4+2 & 1-1 & -3+0 & 1+5 \\ 2+4 & 4+4 & 3-2 & -2+0 \\ 1+1 & 0+3 & 1+1 & 10+5 \end{array}\right] = \left[\begin{array}{cccc} 6 & 0 & -3 & 6 \\ 6 & 8 & 1 & -2 \\ 2 & 3 & 2 & 15 \end{array}\right] \] Next, we take the transpose of the resultant matrix: \[ (M + N)^{\mathrm{T}} = \left[\begin{array}{ccc} 6 & 6 & 2 \\ 0 & 8 & 3 \\ -3 & 1 & 2 \\ 6 & -2 & 15 \end{array}\right] \] 2. Now, let's find \( MN \). To do this, we perform matrix multiplication, which involves the dot product of the rows of \( M \) with the columns of \( N \). Calculating each entry: - First row, first column: \( 4 \times 2 + 1 \times 4 + (-3) \times 1 + 1 \times 5 = 8 + 4 - 3 + 5 = 14 \) - First row, second column: \( 4 \times (-1) + 1 \times 4 + (-3) \times 3 + 1 \times 0 = -4 + 4 - 9 + 0 = -9 \) - First row, third column: \( 4 \times 0 + 1 \times (-2) + (-3) \times 1 + 1 \times 1 = 0 - 2 - 3 + 1 = -4 \) - First row, fourth column: \( 4 \times 5 + 1 \times 0 + (-3) \times 5 + 1 \times 0 = 20 + 0 - 15 + 0 = 5 \) Repeating this process for the second and third rows: Second row, columns: - \( 2 \times 2 + 4 \times 4 + 3 \times 1 + (-2) \times 5 = 4 + 16 + 3 - 10 = 13 \) - \( 2 \times (-1) + 4 \times 4 + 3 \times 3 + (-2) \times 0 = -2 + 16 + 9 + 0 = 23 \) - \( 2 \times 0 + 4 \times (-2) + 3 \times 1 + (-2) \times 1 = 0 - 8 + 3 - 2 = -7 \) - \( 2 \times 5 + 4 \times 0 + 3 \times 5 + (-2) \times 0 = 10 + 0 + 15 + 0 = 25 \) Third row, columns: - \( 1 \times 2 + 0 \times 4 + 1 \times 1 + 10 \times 5 = 2 + 0 + 1 + 50 = 53 \) - \( 1 \times (-1) + 0 \times 4 + 1 \times 3 + 10 \times 0 = -1 + 0 + 3 + 0 = 2 \) - \( 1 \times 0 + 0 \times (-2) + 1 \times 1 + 10 \times 1 = 0 + 0 + 1 + 10 = 11 \) - \( 1 \times 5 + 0 \times 0 + 1 \times 5 + 10 \times 0 = 5 + 0 + 5 + 0 = 10 \) Thus, \[ MN = \left[\begin{array}{cccc} 14 & -9 & -4 & 5 \\ 13 & 23 & -7 & 25 \\ 53 & 2 & 11 & 10 \end{array}\right] \] Now, we take the transpose of this resulting product: \[ (MN)^{\mathrm{T