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

To solve for \(P + 2Q\), you need to apply both scalar multiplication and matrix addition.

Step 1: Multiply \(Q\) by 2
This means multiplying every entry of matrix \(Q\) by 2: \[ Q = \begin{pmatrix} 3 & -7 \\ 1 & 2 \end{pmatrix} \implies 2Q = \begin{pmatrix} 2 \times 3 & 2 \times -7 \\ 2 \times 1 & 2 \times 2 \end{pmatrix} = \begin{pmatrix} 6 & -14 \\ 2 & 4 \end{pmatrix} \]

Step 2: Add \(P\) and \(2Q\)
Now add the corresponding entries of \(P\) and \(2Q\): \[ \begin{align*} P &= \begin{pmatrix} 1 & 3 \\ 2 & -5 \end{pmatrix} \\ 2Q &= \begin{pmatrix} 6 & -14 \\ 2 & 4 \end{pmatrix} \end{align*} \] Add each entry:

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

Underlying Concepts:

• Scalar multiplication: Multiply every element in the matrix by the scalar.
• Matrix addition: Add matrices by adding their corresponding elements.

The answer is the matrix with entries \(7\), \(-11\), \(4\), and \(-1\) as shown above.