Given that $P = \begin{pmatrix} 1 & 3 \\ 2 & -5 \end{pmatrix}$ and Q = $\begin{pmatrix} 3 & -7 \\ 1 & 2 \end{pmatrix}$ . Find P + 2Q

Analysis

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

1. Scalar Multiplication: Multiply every element in matrix $Q$ by 2.
2. Matrix Addition: Add the resulting matrix ($2Q$) to matrix $P$ by summing the corresponding elements.

Concept

If $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and $B = \begin{pmatrix} e & f \\ g & h \end{pmatrix}$, then $A + B = \begin{pmatrix} a+e & b+f \\ c+g & d+h \end{pmatrix}$. Scalar multiplication $kQ$ means multiplying every entry in $Q$ by $k$.

Step-by-Step Solution

Step 1: Calculate $2Q$ Given $Q = \begin{pmatrix} 3 & -7 \\ 1 & 2 \end{pmatrix}$, multiply each element by 2: $2Q = 2 \times \begin{pmatrix} 3 & -7 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 2(3) & 2(-7) \\ 2(1) & 2(2) \end{pmatrix} = \begin{pmatrix} 6 & -14 \\ 2 & 4 \end{pmatrix}$

Step 2: Calculate $P + 2Q$ Now, add matrix $P$ to the result of $2Q$: $P + 2Q = \begin{pmatrix} 1 & 3 \\ 2 & -5 \end{pmatrix} + \begin{pmatrix} 6 & -14 \\ 2 & 4 \end{pmatrix}$

Perform the addition element by element:

• Top-left: $1 + 6 = 7$
• Top-right: $3 + (-14) = 3 - 14 = -11$
• Bottom-left: $2 + 2 = 4$
• Bottom-right: $-5 + 4 = -1$

Resulting matrix: $\begin{pmatrix} 7 & -11 \\ 4 & -1 \end{pmatrix}$

Conclusion

Comparing our result to the given options, we find that it matches option A.

Exam Tip

When dealing with matrix addition or subtraction, always double-check your signs, especially when dealing with negative numbers (like $-5 + 4$). A common mistake is to accidentally add the absolute values instead of performing the algebraic sum. Take your time with the arithmetic!