If A = $\begin{pmatrix}3 & 2 & 1\\4 & 2 & -1\end{pmatrix}$ and B = $\begin{pmatrix}1 & 4\\0 & 1\\3 & 2\end{pmatrix}$. Find A$^T$ + B, ( where T means transpose)

To solve this problem, we need to perform two operations: finding the transpose of matrix A and then adding it to matrix B.

1. Analyze the Question

• Matrix A is a $2 \times 3$ matrix: $\begin{pmatrix}3 & 2 & 1\\4 & 2 & -1\end{pmatrix}$
• Matrix B is a $3 \times 2$ matrix: $\begin{pmatrix}1 & 4\\0 & 1\\3 & 2\end{pmatrix}$
• We need to calculate $A^T + B$.

2. Explain the Concept

• Transpose ($A^T$): To find the transpose of a matrix, you swap its rows and columns. The first row of A becomes the first column of $A^T$, and the second row of A becomes the second column of $A^T$.
• Matrix Addition: You can only add matrices of the same dimensions. Since $A$ is $2 \times 3$, $A^T$ will be $3 \times 2$. Since $B$ is also $3 \times 2$, we can add them by summing their corresponding elements.

3. Show the Solution

Step 1: Find $A^T$ Given $A = \begin{pmatrix}3 & 2 & 1\\4 & 2 & -1\end{pmatrix}$ Swap rows for columns: $A^T = \begin{pmatrix}3 & 4\\2 & 2\\1 & -1\end{pmatrix}$

Step 2: Add $A^T$ and B $A^T + B = \begin{pmatrix}3 & 4\\2 & 2\\1 & -1\end{pmatrix} + \begin{pmatrix}1 & 4\\0 & 1\\3 & 2\end{pmatrix}$

Perform the addition element by element:

• Row 1: $(3+1) = 4$ and $(4+4) = 8$
• Row 2: $(2+0) = 2$ and $(2+1) = 3$
• Row 3: $(1+3) = 4$ and $(-1+2) = 1$

Resulting matrix = $\begin{pmatrix}4 & 8\\2 & 3\\4 & 1\end{pmatrix}$

4. Identify the Answer

Comparing our result with the given options, the correct matrix matches option D.

5. Exam Tip

When dealing with matrix transposes, always double-check your dimensions. If you start with a $2 \times 3$ matrix, its transpose must be $3 \times 2$. If the dimensions don't match the matrix you are adding it to, you have likely made a mistake in the transpose step!