To solve question 2, we need to find $AB$ and $BA$ if possible. Let’s start with the matrices given in the image for question 2:

Matrix $A$: $A = \begin{bmatrix} 2 & 9 \\ -7 & 3 \end{bmatrix}$

Matrix $B$: $B = \begin{bmatrix} 6 & -4 \\ 0 & 3 \end{bmatrix}$

Step 1: Find $AB$

To find $AB$, we multiply each element of the rows in $A$ by the corresponding elements of the columns in $B$:

$AB = \begin{bmatrix} 2 & 9 \\ -7 & 3 \end{bmatrix} \cdot \begin{bmatrix} 6 & -4 \\ 0 & 3 \end{bmatrix}$

Calculations for $AB$:

• The element in the first row, first column: $(2 \cdot 6) + (9 \cdot 0) = 12 + 0 = 12$

• The element in the first row, second column: $(2 \cdot -4) + (9 \cdot 3) = -8 + 27 = 19$

• The element in the second row, first column: $(-7 \cdot 6) + (3 \cdot 0) = -42 + 0 = -42$

• The element in the second row, second column: $(-7 \cdot -4) + (3 \cdot 3) = 28 + 9 = 37$

So, we get: $AB = \begin{bmatrix} 12 & 19 \\ -42 & 37 \end{bmatrix}$

Step 2: Find $BA$

Now we’ll find $BA$, which requires multiplying $B$ by $A$:

$BA = \begin{bmatrix} 6 & -4 \\ 0 & 3 \end{bmatrix} \cdot \begin{bmatrix} 2 & 9 \\ -7 & 3 \end{bmatrix}$

Calculations for $BA$:

• The element in the first row, first column: $(6 \cdot 2) + (-4 \cdot -7) = 12 + 28 = 40$

• The element in the first row, second column: $(6 \cdot 9) + (-4 \cdot 3) = 54 - 12 = 42$

• The element in the second row, first column: $(0 \cdot 2) + (3 \cdot -7) = 0 - 21 = -21$

• The element in the second row, second column: $(0 \cdot 9) + (3 \cdot 3) = 0 + 9 = 9$

So, we get: $BA = \begin{bmatrix} 40 & 42 \\ -21 & 9 \end{bmatrix}$

Summary of Results

• $AB = \begin{bmatrix} 12 & 19 \\ -42 & 37 \end{bmatrix}$
• $BA = \begin{bmatrix} 40 & 42 \\ -21 & 9 \end{bmatrix}$

Would you like further details or explanations?

Here are some additional questions you might find helpful:

1. What are the properties of matrix multiplication that affect the result of $AB$ and $BA$?
2. How do we determine whether two matrices can be multiplied?
3. Is matrix multiplication commutative, and what implications does this have on $AB$ and $BA$?
4. How would the result change if one of the matrices was a different size?
5. Can we generalize a rule for determining if $AB = BA$ for any matrices $A$ and $B$?

Tip: Matrix multiplication requires that the number of columns in the first matrix matches the number of rows in the second matrix.