Let A = $\begin{pmatrix}2 & -4 & 3\\5 & 1 & 0\end{pmatrix}$ and B =  $\begin{pmatrix}1 & 4 & -2\\-3 & 3 & -1\end{pmatrix}$. Find A + 2B

Analysis

To solve this problem, we need to perform two matrix operations:

1. Scalar Multiplication: Multiply every element in matrix B by 2.
2. Matrix Addition: Add the resulting matrix (2B) to matrix A.

Concept

• Scalar Multiplication: If $k$ is a scalar and $B$ is a matrix, $kB$ is found by multiplying each individual entry in $B$ by $k$.
• Matrix Addition: Two matrices can only be added if they have the same dimensions. You add them by summing the corresponding elements (the element in row $i$, column $j$ of matrix A is added to the element in row $i$, column $j$ of matrix 2B).

Step-by-Step Solution

Step 1: Calculate 2B Given $B = \begin{pmatrix}1 & 4 & -2 \\ -3 & 3 & -1\end{pmatrix}$, multiply each element by 2: $2B = 2 \times \begin{pmatrix}1 & 4 & -2 \\ -3 & 3 & -1\end{pmatrix} = \begin{pmatrix}2(1) & 2(4) & 2(-2) \\ 2(-3) & 2(3) & 2(-1)\end{pmatrix}$ $2B = \begin{pmatrix}2 & 8 & -4 \\ -6 & 6 & -2\end{pmatrix}$

Step 2: Calculate A + 2B Now, add matrix A to the result of 2B: $A + 2B = \begin{pmatrix}2 & -4 & 3 \\ 5 & 1 & 0\end{pmatrix} + \begin{pmatrix}2 & 8 & -4 \\ -6 & 6 & -2\end{pmatrix}$

Perform the addition element by element:

• Row 1, Col 1: $2 + 2 = 4$
• Row 1, Col 2: $-4 + 8 = 4$
• Row 1, Col 3: $3 + (-4) = -1$
• Row 2, Col 1: $5 + (-6) = -1$
• Row 2, Col 2: $1 + 6 = 7$
• Row 2, Col 3: $0 + (-2) = -2$

Resulting matrix: $\begin{pmatrix}4 & 4 & -1 \\ -1 & 7 & -2\end{pmatrix}$

Conclusion

Comparing our result to the given options, it matches Option A.

Tip

When dealing with matrix addition or subtraction, always double-check your signs, especially when adding negative numbers. A common mistake is to add the absolute values instead of performing the algebraic sum (e.g., $3 + (-4)$ is $-1$, not $7$).