Math Snap

STEP 1

1. A is a 3x2 matrix and B is a 2x2 matrix.
2. The product AB is given partially, with some missing entries.
3. Matrix multiplication rules apply.

STEP 2

1. Verify if matrix multiplication is possible.
2. Calculate the product AB.
3. Compare the calculated result with the given partial result.
4. Fill in the missing entries if possible.

STEP 3

To multiply matrices A and B, the number of columns in A must equal the number of rows in B.
A is a 3x2 matrix and B is a 2x2 matrix.
A has 2 columns and B has 2 rows, so multiplication is possible.

STEP 4

Let's calculate AB:
\[AB = \begin{bmatrix} -1 & 9 \\ -4 & 5 \\ 0 & 4 \end{bmatrix} \times
\begin{bmatrix} 3 & 4 \\ 0 & 9 \end{bmatrix}\] Calculating each element:
$$AB_{11} = (-1)(3) + (9)(0) = -3$$ $$AB_{12} = (-1)(4) + (9)(9) = 77$$ $$AB_{21} = (-4)(3) + (5)(0) = -12$$ $$AB_{22} = (-4)(4) + (5)(9) = 29$$ $$AB_{31} = (0)(3) + (4)(0) = 0$$ $$AB_{32} = (0)(4) + (4)(9) = 36$$ Therefore,
$$AB = \begin{bmatrix} -3 & 77 \\ -12 & 29 \\ 0 & 36 \end{bmatrix}$$

STEP 5

Now, let's compare our calculated result with the given partial result:
Calculated:
$$AB = \begin{bmatrix} -3 & 77 \\ -12 & 29 \\ 0 & 36 \end{bmatrix}$$ Given:
$$AB = \begin{bmatrix} ? & 77 \\ -12 & 29 \\ ? & ? \end{bmatrix}$$ We can see that the entries that are given match our calculated results.

We can now fill in the missing entries:
$$AB = \begin{bmatrix} -3 & 77 \\ -12 & 29 \\ 0 & 36 \end{bmatrix}$$ Therefore, the complete product AB is:
$$AB = \begin{bmatrix} -3 & 77 \\ -12 & 29 \\ 0 & 36 \end{bmatrix}$$