If P = $\begin{pmatrix} 5 & 3 \\ 2 & 1 \end{pmatrix}$ and Q = $\begin{pmatrix} 4 & 2 \\ 3 & 5 \end{pmatrix}$, find 2P + Q

To solve the expression 2P + Q, we need to perform two operations: scalar multiplication and matrix addition.

1. Analyze the question

We are given two $2 \times 2$ matrices: $P = \begin{pmatrix} 5 & 3 \\ 2 & 1 \end{pmatrix}$ and $Q = \begin{pmatrix} 4 & 2 \\ 3 & 5 \end{pmatrix}$ We need to find the result of $2P + Q$.

2. Explain the concept

• Scalar Multiplication: To multiply a matrix by a scalar (in this case, 2), you multiply every individual element inside the matrix by that number.
• Matrix Addition: To add two matrices of the same order, you add the corresponding elements (the element in the first row/first column of the first matrix is added to the element in the first row/first column of the second matrix, and so on).

3. Show the solution

Step 1: Calculate 2P Multiply every element in matrix P by 2: $2P = 2 \times \begin{pmatrix} 5 & 3 \\ 2 & 1 \end{pmatrix} = \begin{pmatrix} 2 \times 5 & 2 \times 3 \\ 2 \times 2 & 2 \times 1 \end{pmatrix} = \begin{pmatrix} 10 & 6 \\ 4 & 2 \end{pmatrix}$

Step 2: Add Q to the result Now, add matrix Q to the matrix we just calculated: $2P + Q = \begin{pmatrix} 10 & 6 \\ 4 & 2 \end{pmatrix} + \begin{pmatrix} 4 & 2 \\ 3 & 5 \end{pmatrix}$

Perform the addition element-wise: $= \begin{pmatrix} 10 + 4 & 6 + 2 \\ 4 + 3 & 2 + 5 \end{pmatrix}$ $= \begin{pmatrix} 14 & 8 \\ 7 & 7 \end{pmatrix}$

4. Identify the answer

Comparing our result $\begin{pmatrix} 14 & 8 \\ 7 & 7 \end{pmatrix}$ with the given options, it matches option B.

5. Exam Tip

When dealing with matrix operations in JAMB, always double-check your arithmetic for the individual elements. A common mistake is to add the matrices before multiplying by the scalar. Remember the BODMAS rule: perform the multiplication (scalar) before the addition. Also, ensure you don't confuse the row and column positions when adding!