Addition of Matrices: Calculate $D + E$ .D = \left[\begin{array}{ccc}1 & 5 & 2\-1 & 0 & 1\3 & 2 & 4\end{array}\right]E = \left[\begin{array}{ccc}6 & 1 & 3\-1 & 1 & 2\4 & 1 & 3\end{array}\right]To add two matrices, we add the corresponding elements.D + E = \left[\begin{array}{ccc}1+6 & 5+1 & 2+3\-1-1 & 0+1 & 1+2\3+4 & 2+1 & 4+3\end{array}\right]D + E = \left[\begin{array}{ccc}7 & 6 & 5\-2 & 1 & 3\7 & 3 & 7\end{array}\right]
Subtraction of Matrices: Calculate $D - E$ .
$D - E = [[1-6, 5-1, 2-3], [-1-(-1), 0-1, 1-2], [3-4, 2-1, 4-3]]$
$D - E = [[-5, 4, -1], [0, -1, -1], [-1, 1, 1]]$
Scalar Multiplication: Calculate $5A$ . $A = \left[\begin{array}{cc}3 & 0 \ -1 & 2 \ 1 & 1\end{array}\right]$ To multiply a matrix by a scalar, we multiply each element by the scalar. $5A = \left[\begin{array}{cc}5\times 3 & 5\times 0 \ 5\times (-1) & 5\times 2 \ 5\times 1 & 5\times 1\end{array}\right]$ $5A = \left[\begin{array}{cc}15 & 0 \ -5 & 10 \ 5 & 5\end{array}\right]$
Scalar Multiplication: Calculate -7C＂).$\newline\C = \left[\begin{array}{ccc} 1 & 4 & 2 \ 3 & 1 & 5 \end{array}\right]$$\newline$$-7C = \left[\begin{array}{ccc} -7\times 1 & -7\times 4 & -7\times 2 \ -7\times 3 & -7\times 1 & -7\times 5 \end{array}\right]$$\newline$$-7C = \left[\begin{array}{ccc} -7 & -28 & -14 \ -21 & -7 & -35 \end{array}\right]$
Incompatible Matrices: Calculate $2B - C$.$B = \left[\begin{array}{cc}4 & -1 \ 0 & 2\end{array}\right]$$C = \left[\begin{array}{ccc}1 & 4 & 2 \ 3 & 1 & 5\end{array}\right]$Since $B$ is a $2 \times 2$ matrix and $C$ is a $2 \times 3$ matrix, we cannot perform this operation because the matrices are not of the same size.
Matrix Operations: Calculate $4E - 2D$.$\newline$E = \begin{bmatrix} $6$ & $1$ & $3$ \ $-1$ & $1$ & $2$ \ $4$ & $1$ & $3$ \end{bmatrix}, $\newline$ D = \begin{bmatrix} $1$ & $5$ & $2$ \ $-1$ & $0$ & $1$ \ $3$ & $2$ & $4$ \end{bmatrix}$$\newline$$4E - 2D = \begin{bmatrix} 4\times 6 - 2\times 1 & 4\times 1 - 2\times 5 & 4\times 3 - 2\times 2 \ 4\times (-1) - 2\times (-1) & 4\times 1 - 2\times 0 & 4\times 2 - 2\times 1 \ 4\times 4 - 2\times 3 & 4\times 1 - 2\times 2 & 4\times 3 - 2\times 4 \end{bmatrix}$$\newline$$4E - 2D = \begin{bmatrix} 24 - 2 & 4 - 10 & 12 - 4 \ -4 + 2 & 4 - 0 & 8 - 2 \ 16 - 6 & 4 - 4 & 12 - 8 \end{bmatrix}$$\newline$$4E - 2D = \begin{bmatrix} 22 & -6 & 8 \ -2 & 4 & 6 \ 10 & 0 & 4 \end{bmatrix}$
Scalar Multiplication: Calculate $-3(D + 2E)$.$\newline$First, we calculate $D + 2E$.$\newline$$D + 2E = [[1 + 2\times6, 5 + 2\times1, 2 + 2\times3], [-1 + 2\times(-1), 0 + 2\times1, 1 + 2\times2], [3 + 2\times4, 2 + 2\times1, 4 + 2\times3]]$$\newline$$D + 2E = [[1 + 12, 5 + 2, 2 + 6], [-1 - 2, 0 + 2, 1 + 4], [3 + 8, 2 + 2, 4 + 6]]$$\newline$$D + 2E = [[13, 7, 8], [-3, 2, 5], [11, 4, 10]]$$\newline$Now, we multiply by $-3$.$\newline$$-3(D + 2E) = [[-3\times13, -3\times7, -3\times8], [-3\times(-3), -3\times2, -3\times5], [-3\times11, -3\times4, -3\times10]]$$\newline$$-3(D + 2E) = [[-39, -21, -24], [9, -6, -15], [-33, -12, -30]]$
Trace Calculation: Calculate $A - A$.$A = \left[\begin{array}{cc}3 & 0 \-1 & 2 \1 & 1\end{array}\right]$$A - A = \left[\begin{array}{cc}3-3 & 0-0 \-1-(-1) & 2-2 \1-1 & 1-1\end{array}\right]$$A - A = \left[\begin{array}{cc}0 & 0 \0 & 0 \0 & 0\end{array}\right]$
Trace Calculation: Calculate $tr(D)$.$D = \left[\begin{array}{ccc}1 & 5 & 2 \-1 & 0 & 1 \3 & 2 & 4\end{array}\right]$The trace of a matrix is the sum of the elements on the main diagonal.$tr(D) = 1 + 0 + 4$$tr(D) = 5$
Scalar Multiplication: Calculate $tr(D - 3E)$. First, we calculate $D - 3E$. $D - 3E = [[1 - 3\times6, 5 - 3\times1, 2 - 3\times3], [-1 - 3\times(-1), 0 - 3\times1, 1 - 3\times2], [3 - 3\times4, 2 - 3\times1, 4 - 3\times3]]$ $D - 3E = [[1 - 18, 5 - 3, 2 - 9], [-1 + 3, 0 - 3, 1 - 6], [3 - 12, 2 - 3, 4 - 9]]$ $D - 3E = [[-17, 2, -7], [2, -3, -5], [-9, -1, -5]]$ Now, we calculate the trace. $tr(D - 3E) = -17 - 3 - 5$ $tr(D - 3E) = -25$
Trace Calculation: Calculate $4\text{tr}(7B)$.
$B = \begin{bmatrix} 4 & -1 \ 0 & 2 \end{bmatrix}$
First, we calculate the trace of $B$.
$\text{tr}(B) = 4 + 2$
$\text{tr}(B) = 6$
Now, we multiply by $7$ and then by $4$.
$4\text{tr}(7B) = 4 \times 7 \times \text{tr}(B)$
$4\text{tr}(7B) = 4 \times 7 \times 6$
$4\text{tr}(7B) = 168$
Trace Calculation: Calculate $4\text{tr}(7B)$.
$B = \begin{bmatrix} 4 & -1 \ 0 & 2 \end{bmatrix}$
First, we calculate the trace of $B$.
$\text{tr}(B) = 4 + 2$
$\text{tr}(B) = 6$
Now, we multiply by $7$ and then by $4$.
$4\text{tr}(7B) = 4 \times 7 \times \text{tr}(B)$
$4\text{tr}(7B) = 4 \times 7 \times 6$
$4\text{tr}(7B) = 168$
Calculate $B = \begin{bmatrix} 4 & -1 \ 0 & 2 \end{bmatrix}$$0$.
$B = \begin{bmatrix} 4 & -1 \ 0 & 2 \end{bmatrix}$$1$
Since $B = \begin{bmatrix} 4 & -1 \ 0 & 2 \end{bmatrix}$$2$ is not a square matrix, we cannot calculate the trace of $B = \begin{bmatrix} 4 & -1 \ 0 & 2 \end{bmatrix}$$2$. The trace is only defined for square matrices.