If A=`[{:(,1,2),(,3,-4),(,5,6):}]and B=[{:(,4,5,6),(,7,-8,2):}]` will AB be equal to BA. Also find AB & BA.

Answer

To determine whether \( AB \) is equal to \( BA \) for the given matrices \( A \) and \( B \), we first need to calculate both products.

### Given Matrices:
\[ A = \begin{bmatrix} 1 & 2 & 3 \\ -4 & 5 & 6 \end{bmatrix} \]
\[ B = \begin{bmatrix} 4 & 5 & 6 \\ 7 & -8 & 2 \end{bmatrix} \]

### Step 1: Calculate \( AB \)

To find \( AB \), we multiply matrix \( A \) by matrix \( B \). The resulting matrix will be a \( 2 \times 3 \) matrix.

\[ AB = \begin{bmatrix} 1 & 2 & 3 \\ -4 & 5 & 6 \end{bmatrix} \begin{bmatrix} 4 & 5 & 6 \\ 7 & -8 & 2 \end{bmatrix} \]

Calculating each element of \( AB \):

- First row, first column:
\[ 1 \cdot 4 + 2 \cdot 7 = 4 + 14 = 18 \]

- First row, second column:
\[ 1 \cdot 5 + 2 \cdot (-8) = 5 - 16 = -11 \]

- First row, third column:
\[ 1 \cdot 6 + 2 \cdot 2 = 6 + 4 = 10 \]

- Second row, first column:
\[ -4 \cdot 4 + 5 \cdot 7 = -16 + 35 = 19 \]

- Second row, second column:
\[ -4 \cdot 5 + 5 \cdot (-8) = -20 - 40 = -60 \]

- Second row, third column:
\[ -4 \cdot 6 + 5 \cdot 2 = -24 + 10 = -14 \]

Thus, we have:
\[ AB = \begin{bmatrix} 18 & -11 & 10 \\ 19 & -60 & -14 \end{bmatrix} \]

### Step 2: Calculate \( BA \)

Now, we calculate \( BA \) by multiplying matrix \( B \) by matrix \( A \). The resulting matrix will be a \( 3 \times 2 \) matrix.

\[ BA = \begin{bmatrix} 4 & 5 & 6 \\ 7 & -8 & 2 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \\ -4 & 5 & 6 \end{bmatrix} \]

Calculating each element of \( BA \):

- First row, first column:
\[ 4 \cdot 1 + 5 \cdot (-4) = 4 - 20 = -16 \]

- First row, second column:
\[ 4 \cdot 2 + 5 \cdot 5 = 8 + 25 = 33 \]

- Second row, first column:
\[ 7 \cdot 1 + (-8) \cdot (-4) = 7 + 32 = 39 \]

- Second row, second column:
\[ 7 \cdot 2 + (-8) \cdot 5 = 14 - 40 = -26 \]

- Third row, first column:
\[ 6 \cdot 1 + 2 \cdot (-4) = 6 - 8 = -2 \]

- Third row, second column:
\[ 6 \cdot 2 + 2 \cdot 5 = 12 + 10 = 22 \]

Thus, we have:
\[ BA = \begin{bmatrix} -16 & 33 \\ 39 & -26 \\ -2 & 22 \end{bmatrix} \]

### Conclusion

Now we can summarize the results:
\[ AB = \begin{bmatrix} 18 & -11 & 10 \\ 19 & -60 & -14 \end{bmatrix} \]
\[ BA = \begin{bmatrix} -16 & 33 \\ 39 & -26 \\ -2 & 22 \end{bmatrix} \]

Since \( AB \) is a \( 2 \times 3 \) matrix and \( BA \) is a \( 3 \times 2 \) matrix, they cannot be equal. Therefore, \( AB \neq BA \).