Answer
$B^3$ is a defined expression, and the resulting matrix is:
$B^3 =|1 \space \space -3| $
$ \hphantom {B^3 =.} | 0 \hphantom {B...} 1| $
Work Step by Step
** As we have calculated on the last exercise:
$B^2 =| 1 \space \space -2| $
$ \hphantom {B^2 =.} | 0 \hphantom {--.}1| $
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1. Check if the expression is defined.
$B^3$ is equal to $BB^2$ (or $B^2B$)(both are the same).
For a matrix multiplication, the length of the row in the first matrix must be equal to the length of the column in the second one.
$B$ row length: $2$
$B^2$ column length: $2$
Therefore, the expression is defined and the multiplication is possible.
2. Evaluate it:
$(B^3)_{11} = (B_{11} \times B^2_{11} + B_{12} \times B^2_{21})$
$(B^3)_{12} = (B_{11} \times B^2_{12} + B_{12} \times B^2_{22})$
$(B^3)_{21} = (B_{21} \times B^2_{11} + B_{22} \times B^2_{21})$
$(B^3)_{22} = (B_{21} \times B^2_{12} + B_{22} \times B^2_{22})$
$(B^3)_{11} = 1 \times 1 + (-1) \times 0 = 1$
$(B^3)_{12} = 1 \times (-2) + (-1) \times 1 = -3$
$(B^3)_{21} = 0 \times 1 + 1 \times 0 = 0$
$(B^3)_{22} = 0 \times (-2) + 1 \times 1 = 1$
$B^3 =|1 \space \space -3| $
$ \hphantom {B^3 =.} | 0 \hphantom {B...} 1| $
