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Math  /  Algebra

QuestionIf possible, find ABA B. (If not possible, enter IMPOSSIBLE in any cell of the matrix.) A=[−19−4504],B=[3409]AB=−7c−1277⇒−39\begin{array}{c} A=\left[\begin{array}{rr} -1 & 9 \\ -4 & 5 \\ 0 & 4 \end{array}\right], \quad B=\left[\begin{array}{ll} 3 & 4 \\ 0 & 9 \end{array}\right] \\ A B=-7 c \\ -12 \end{array} \begin{array}{cc} 77 & \Rightarrow \\ -39 \end{array} -12 □\square 77 □\square 29

Studdy Solution

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=[−19−4504]×[3409]AB = \begin{bmatrix} -1 & 9 \\ -4 & 5 \\ 0 & 4 \end{bmatrix} \times \begin{bmatrix} 3 & 4 \\ 0 & 9 \end{bmatrix}
Calculating each element:
AB11=(−1)(3)+(9)(0)=−3AB_{11} = (-1)(3) + (9)(0) = -3 AB12=(−1)(4)+(9)(9)=77AB_{12} = (-1)(4) + (9)(9) = 77 AB21=(−4)(3)+(5)(0)=−12AB_{21} = (-4)(3) + (5)(0) = -12 AB22=(−4)(4)+(5)(9)=29AB_{22} = (-4)(4) + (5)(9) = 29 AB31=(0)(3)+(4)(0)=0AB_{31} = (0)(3) + (4)(0) = 0 AB32=(0)(4)+(4)(9)=36AB_{32} = (0)(4) + (4)(9) = 36
Therefore,
AB=[−377−1229036]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=[−377−1229036]AB = \begin{bmatrix} -3 & 77 \\ -12 & 29 \\ 0 & 36 \end{bmatrix}
Given: AB=[?77−1229??]AB = \begin{bmatrix} ? & 77 \\ -12 & 29 \\ ? & ? \end{bmatrix}
We can see that the entries that are given match our calculated results.

STEP 6

We can now fill in the missing entries:
AB=[−377−1229036]AB = \begin{bmatrix} -3 & 77 \\ -12 & 29 \\ 0 & 36 \end{bmatrix}
Therefore, the complete product AB is:
AB=[−377−1229036]AB = \begin{bmatrix} -3 & 77 \\ -12 & 29 \\ 0 & 36 \end{bmatrix}

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