If `P =[(3,4),(-1,2),(0,1)] and Q = [(-1, 2,1),(1,2,3)]`, then `(P^(T) + Q)=`

Options

A. `[(2,5),(1,4),(1,4)]`
B. `[(4,7),(1,3),(9,2)]`
C. `[(2,1,1),(5,4,4)]`
D. `[(4,1,9),(7,3,2)]`

Answer

To solve the problem, we need to find \( P^T + Q \), where \( P \) and \( Q \) are given matrices. Let's break it down step by step.

### Step 1: Define the matrices \( P \) and \( Q \)

Given:
\[ P = \begin{pmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{pmatrix} \]
\[ Q = \begin{pmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{pmatrix} \]

### Step 2: Calculate the transpose of matrix \( P \)

The transpose of a matrix is obtained by swapping its rows with columns. Thus, the transpose \( P^T \) will be:
\[ P^T = \begin{pmatrix} 3 & -1 & 0 \\ 4 & 2 & 1 \end{pmatrix} \]

### Step 3: Add matrices \( P^T \) and \( Q \)

Now, we need to add \( P^T \) and \( Q \). Before we proceed, we should confirm that both matrices have the same dimensions.

- \( P^T \) is a \( 2 \times 3 \) matrix.
- \( Q \) is also a \( 2 \times 3 \) matrix.

Since both matrices have the same dimensions, we can add them.

The addition is performed element-wise:
\[ P^T + Q = \begin{pmatrix} 3 & -1 & 0 \\ 4 & 2 & 1 \end{pmatrix} + \begin{pmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{pmatrix} \]

Calculating the sum:
- First row:
- \( 3 + (-1) = 2 \)
- \( -1 + 2 = 1 \)
- \( 0 + 1 = 1 \)

- Second row:
- \( 4 + 1 = 5 \)
- \( 2 + 2 = 4 \)
- \( 1 + 3 = 4 \)

Thus, we have:
\[ P^T + Q = \begin{pmatrix} 2 & 1 & 1 \\ 5 & 4 & 4 \end{pmatrix} \]

### Final Answer:
\[ P^T + Q = \begin{pmatrix} 2 & 1 & 1 \\ 5 & 4 & 4 \end{pmatrix} \]

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