Given matrix M = $\begin{vmatrix} -2 & 0 & 4 \\ 0 & -1 & 6 \\ 5 & 6 & 3 \end{vmatrix}$, find $M^{T} + 2M$

To solve this problem, we need to perform two operations: find the transpose of matrix $M$ ($M^T$) and calculate $2M$, then add them together.

1. Analyze the components

Given $M = \begin{pmatrix} -2 & 0 & 4 \\ 0 & -1 & 6 \\ 5 & 6 & 3 \end{pmatrix}$

Step A: Find the Transpose ($M^T$) To find the transpose, we swap the rows and columns. The first row becomes the first column, the second row becomes the second column, and the third row becomes the third column. $M^T = \begin{pmatrix} -2 & 0 & 5 \\ 0 & -1 & 6 \\ 4 & 6 & 3 \end{pmatrix}$

Step B: Find $2M$ To find $2M$, we multiply every element inside matrix $M$ by 2: $2M = 2 \times \begin{pmatrix} -2 & 0 & 4 \\ 0 & -1 & 6 \\ 5 & 6 & 3 \end{pmatrix} = \begin{pmatrix} -4 & 0 & 8 \\ 0 & -2 & 12 \\ 10 & 12 & 6 \end{pmatrix}$

2. Perform the Addition ($M^T + 2M$)

Now, we add the corresponding elements of $M^T$ and $2M$:

$\begin{pmatrix} -2 & 0 & 5 \\ 0 & -1 & 6 \\ 4 & 6 & 3 \end{pmatrix} + \begin{pmatrix} -4 & 0 & 8 \\ 0 & -2 & 12 \\ 10 & 12 & 6 \end{pmatrix}$

• Row 1: $(-2 + -4) = -6$, $(0 + 0) = 0$, $(5 + 8) = 13$
• Row 2: $(0 + 0) = 0$, $(-1 + -2) = -3$, $(6 + 12) = 18$
• Row 3: $(4 + 10) = 14$, $(6 + 12) = 18$, $(3 + 6) = 9$

Resulting Matrix: $\begin{pmatrix} -6 & 0 & 13 \\ 0 & -3 & 18 \\ 14 & 18 & 9 \end{pmatrix}$

3. Conclusion

Comparing our result to the given options, it matches option B.

Exam Tip: When dealing with matrix operations, always double-check your signs (positive/negative). A common mistake in JAMB is adding a negative number incorrectly (e.g., $-1 + -2$ becoming $+1$ instead of $-3$). Take your time with the arithmetic!