To find [tex]\( A^2 \)[/tex] for a given matrix [tex]\( A \)[/tex], we need to perform matrix multiplication of [tex]\( A \)[/tex] with itself. This operation is only defined for square matrices, i.e., matrices with the same number of rows and columns. Let's go through each of the matrices provided and determine [tex]\( A^2 \)[/tex].
### (i) [tex]\( A = \left[\begin{array}{cc} 3 & 2 \\ 1 & 4 \end{array}\right] \)[/tex]
Matrix [tex]\( A \)[/tex] is a [tex]\( 2 \times 2 \)[/tex] square matrix, so [tex]\( A^2 \)[/tex] is defined.
[tex]\[ A^2 = A \cdot A
\][/tex]
Performing the matrix multiplication:
[tex]\[
A^2 = \left[\begin{array}{cc} 3 & 2 \\ 1 & 4 \end{array}\right] \cdot \left[\begin{array}{cc} 3 & 2 \\ 1 & 4 \end{array}\right]
\][/tex]
[tex]\[ = \left[\begin{array}{cc} 3 \times 3 + 2 \times 1 & 3 \times 2 + 2 \times 4 \\ 1 \times 3 + 4 \times 1 & 1 \times 2 + 4 \times 4 \end{array}\right]
\][/tex]
[tex]\[ = \left[\begin{array}{cc} 9 + 2 & 6 + 8 \\ 3 + 4 & 2 + 16 \end{array}\right]
\][/tex]
[tex]\[ = \left[\begin{array}{cc} 11 & 14 \\ 7 & 18 \end{array}\right]
\][/tex]
### (ii) [tex]\( A = \left[\begin{array}{ccc} 1 & 2 & 3\end{array}\right] \)[/tex]
and
[tex]\( A = \left[\begin{array}{c} 3 \\ 2 \\ 4\end{array}\right] \)[/tex]
Here, we are given two separate matrices. The first is a [tex]\( 1 \times 3 \)[/tex] row vector and the second is a [tex]\( 3 \times 1 \)[/tex] column vector. Neither of these is a square matrix, hence [tex]\( A^2 \)[/tex] is not defined for either of them.
[tex]\[ A^2 \text{ is not defined} \][/tex]
### (iv) [tex]\( A = \left[\begin{array}{ccc}-1 & 1 & 2 \\ 3 & 1 & 3 \\ 0 & 1 & 2\end{array}\right] \)[/tex]
Matrix [tex]\( A \)[/tex] is a [tex]\( 3 \times 3 \)[/tex] square matrix, so [tex]\( A^2 \)[/tex] is defined.
[tex]\[ A^2 = A \cdot A
\][/tex]
Performing the matrix multiplication:
[tex]\[
A^2 = \left[\begin{array}{ccc}-1 & 1 & 2 \\ 3 & 1 & 3 \\ 0 & 1 & 2\end{array}\right] \cdot \left[\begin{array}{ccc}-1 & 1 & 2 \\ 3 & 1 & 3 \\ 0 & 1 & 2\end{array}\right]
\][/tex]
[tex]\[ = \left[\begin{array}{ccc}(-1 \times -1) + (1 \times 3) + (2 \times 0) & (-1 \times 1) + (1 \times 1) + (2 \times 1) & (-1 \times 2) + (1 \times 3) + (2 \times 2) \\ (3 \times -1) + (1 \times 3) + (3 \times 0) & (3 \times 1) + (1 \times 1) + (3 \times 1) & (3 \times 2) + (1 \times 3) + (3 \times 2) \\ (0 \times -1) + (1 \times 3) + (2 \times 0) & (0 \times 1) + (1 \times 1) + (2 \times 1) & (0 \times 2) + (1 \times 3) + (2 \times 2) \end{array}\right]
\][/tex]
[tex]\[ = \left[\begin{array}{ccc}1 + 3 + 0 & -1 + 1 + 2 & -2 + 3 + 4 \\ -3 + 3 + 0 & 3 + 1 + 3 & 6 + 3 + 6 \\ 0 + 3 + 0 & 0 + 1 + 2 & 0 + 3 + 4 \end{array}\right]
\][/tex]
[tex]\[ = \left[\begin{array}{ccc} 4 & 2 & 5 \\ 0 & 7 & 15 \\ 3 & 3 & 7 \end{array}\right]
\][/tex]
### (v) [tex]\( A = \left[\begin{array}{ccc}1 & 2 & 3 \\ 3 & 2 & 1 \end{array}\right] \)[/tex]
Matrix [tex]\( A \)[/tex] is a [tex]\( 2 \times 3 \)[/tex] matrix, which is not a square matrix, so [tex]\( A^2 \)[/tex] is not defined.
[tex]\[ A^2 \text{ is not defined} \][/tex]
To summarize:
- For matrix (i), [tex]\( A^2 \)[/tex] is [tex]\(\left[\begin{array}{cc} 11 & 14 \\ 7 & 18 \end{array}\right] \)[/tex].
- For matrices (ii) and (iii), [tex]\( A^2 \)[/tex] is not defined.
- For matrix (iv), [tex]\( A^2 \)[/tex] is [tex]\(\left[\begin{array}{ccc} 4 & 2 & 5 \\ 0 & 7 & 15 \\ 3 & 3 & 7 \end{array}\right] \)[/tex].
- For matrix (v), [tex]\( A^2 \)[/tex] is not defined.
