QuestionIf possible, find $A B, B A$ , and $A^{2}$ . (If not possible, enter IMPOSSIBLE in any cell of the matrix.) $A=\left[\begin{array}{llll} 1 & 2 & 4 & 5 \end{array}\right], \quad B=\left[\begin{array}{l} 3 \\ 4 \\ 0 \\ 1 \end{array}\right]$ (a) $A B$ (b) $B A$

Studdy Solution

STEP 1

1. Matrix multiplication is defined if the number of columns in the first matrix equals the number of rows in the second matrix.
2. The matrix $A$ is a 1x4 matrix (1 row, 4 columns).
3. The matrix $B$ is a 4x1 matrix (4 rows, 1 column).
4. To find $A^2$ , $A$ must be a square matrix, which it is not.

STEP 2

1. Determine if $A B$ is possible and calculate it if so.
2. Determine if $B A$ is possible and calculate it if so.
3. Determine if $A^2$ is possible.

STEP 3

To determine if $A B$ is possible, check the dimensions: - $A$ is 1x4 - $B$ is 4x1
Since the number of columns in $A$ (4) equals the number of rows in $B$ (4), $A B$ is possible.
Calculate $A B$ :
$A B = \left[\begin{array}{llll} 1 & 2 & 4 & 5 \end{array}\right] \left[\begin{array}{l} 3 \\ 4 \\ 0 \\ 1 \end{array}\right]$
Perform the multiplication:
$A B = (1 \times 3) + (2 \times 4) + (4 \times 0) + (5 \times 1)$ $A B = 3 + 8 + 0 + 5$ $A B = 16$

STEP 4

To determine if $B A$ is possible, check the dimensions: - $B$ is 4x1 - $A$ is 1x4
Since the number of columns in $B$ (1) does not equal the number of rows in $A$ (1), $B A$ is not possible.
$B A = \text{IMPOSSIBLE}$

STEP 5

To determine if $A^2$ is possible, $A$ must be a square matrix. Since $A$ is 1x4, it is not a square matrix.
$A^2 = \text{IMPOSSIBLE}$
The solutions are: (a) $A B = 16$ (b) $B A = \text{IMPOSSIBLE}$ (c) $A^2 = \text{IMPOSSIBLE}$

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