For two matrices $A = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B^T = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}$, A - B equals

Given:

$A = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix}$

$B^T = \begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}$

To find $A - B$, matrix $B$ is needed first, not $B^T$.

To find $B$ from $B^T$, transpose by flipping rows and columns.

The columns of $B^T$ become the rows of $B$:

Column 1 of $B^T$: $\begin{bmatrix} -1 \\ 1 \end{bmatrix}$ becomes Row 1 of $B$

Column 2 of $B^T$: $\begin{bmatrix} 2 \\ 2 \end{bmatrix}$ becomes Row 2 of $B$

Column 3 of $B^T$: $\begin{bmatrix} 1 \\ 3 \end{bmatrix}$ becomes Row 3 of $B$

Therefore:

$B = \begin{bmatrix} -1 & 1 \\ 2 & 2 \\ 1 & 3 \end{bmatrix}$

$A - B = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix} - \begin{bmatrix} -1 & 1 \\ 2 & 2 \\ 1 & 3 \end{bmatrix}$

Subtracting element by element:

Row 1, Col 1: $3 - (-1) = 4$

Row 1, Col 2: $4 - 1 = 3$

Row 2, Col 1: $-1 - 2 = -3$

Row 2, Col 2: $2 - 2 = 0$

Row 3, Col 1: $0 - 1 = -1$

Row 3, Col 2: $1 - 3 = -2$

Therefore:

$A - B = \begin{bmatrix} 4 & 3 \\ -3 & 0 \\ -1 & -2 \end{bmatrix}$