Solved on Dec 09, 2023

Compute the product of two 3x3 matrices: $\left[\begin{array}{ccc}0 & 0 & 4 \\ -2 & -2 & 2 \\ 0 & -2 & -5\end{array}\right]$ and $\left[\begin{array}{ccc}-1 & -1 & -1 \\ 1 & -3 & 0 \\ 0 & 1 & 1\end{array}\right]$ .

STEP 1

Assumptions
1. We are given two matrices to multiply.
2. The first matrix is a 3x3 matrix denoted as $A$ .
3. The second matrix is also a 3x3 matrix denoted as $B$ .
4. Matrix multiplication is defined as $(AB)_{ij} = \sum_{k=1}^{n} A_{ik}B_{kj}$ , where $n$ is the number of columns of $A$ and the number of rows of $B$ (which must be equal).

STEP 2

Write down the given matrices $A$ and $B$ .
$A = \left[\begin{array}{ccc} 0 & 0 & 4 \\ -2 & -2 & 2 \\ 0 & -2 & -5 \end{array}\right], \quad B = \left[\begin{array}{ccc} -1 & -1 & -1 \\ 1 & -3 & 0 \\ 0 & 1 & 1 \end{array}\right]$

STEP 3

We will calculate the product $AB$ by computing each element $(AB)_{ij}$ of the resulting matrix.

STEP 4

Calculate the element $(AB)_{11}$ .
$(AB)_{11} = A_{11}B_{11} + A_{12}B_{21} + A_{13}B_{31} = 0 \cdot (-1) + 0 \cdot 1 + 4 \cdot 0 = 0$

STEP 5

Calculate the element $(AB)_{12}$ .
$(AB)_{12} = A_{11}B_{12} + A_{12}B_{22} + A_{13}B_{32} = 0 \cdot (-1) + 0 \cdot (-3) + 4 \cdot 1 = 4$

STEP 6

Calculate the element $(AB)_{13}$ .
$(AB)_{13} = A_{11}B_{13} + A_{12}B_{23} + A_{13}B_{33} = 0 \cdot (-1) + 0 \cdot 0 + 4 \cdot 1 = 4$

STEP 7

Calculate the element $(AB)_{21}$ .
$(AB)_{21} = A_{21}B_{11} + A_{22}B_{21} + A_{23}B_{31} = (-2) \cdot (-1) + (-2) \cdot 1 + 2 \cdot 0 = 2 - 2 + 0 = 0$

STEP 8

Calculate the element $(AB)_{22}$ .
$(AB)_{22} = A_{21}B_{12} + A_{22}B_{22} + A_{23}B_{32} = (-2) \cdot (-1) + (-2) \cdot (-3) + 2 \cdot 1 = 2 + 6 + 2 = 10$

STEP 9

Calculate the element $(AB)_{23}$ .
$(AB)_{23} = A_{21}B_{13} + A_{22}B_{23} + A_{23}B_{33} = (-2) \cdot (-1) + (-2) \cdot 0 + 2 \cdot 1 = 2 + 0 + 2 = 4$

STEP 10

Calculate the element $(AB)_{31}$ .
$(AB)_{31} = A_{31}B_{11} + A_{32}B_{21} + A_{33}B_{31} = 0 \cdot (-1) + (-2) \cdot 1 + (-5) \cdot 0 = 0 - 2 + 0 = -2$

STEP 11

Calculate the element $(AB)_{32}$ .
$(AB)_{32} = A_{31}B_{12} + A_{32}B_{22} + A_{33}B_{32} = 0 \cdot (-1) + (-2) \cdot (-3) + (-5) \cdot 1 = 0 + 6 - 5 = 1$

STEP 12

Calculate the element $(AB)_{33}$ .
$(AB)_{33} = A_{31}B_{13} + A_{32}B_{23} + A_{33}B_{33} = 0 \cdot (-1) + (-2) \cdot 0 + (-5) \cdot 1 = 0 + 0 - 5 = -5$

STEP 13

Combine all the calculated elements to form the resulting matrix $AB$ .
$AB = \left[\begin{array}{ccc} (AB)_{11} & (AB)_{12} & (AB)_{13} \\ (AB)_{21} & (AB)_{22} & (AB)_{23} \\ (AB)_{31} & (AB)_{32} & (AB)_{33} \end{array}\right]$

STEP 14

Substitute the calculated values into the resulting matrix.
$AB = \left[\begin{array}{ccc} 0 & 4 & 4 \\ 0 & 10 & 4 \\ -2 & 1 & -5 \end{array}\right]$
The product of the given matrices is:
$\left[\begin{array}{ccc} 0 & 4 & 4 \\ 0 & 10 & 4 \\ -2 & 1 & -5 \end{array}\right]$

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Compute the product of two 3x3 matrices: $\left[\begin{array}{ccc}0 & 0 & 4 \\ -2 & -2 & 2 \\ 0 & -2 & -5\end{array}\right]$ and $\left[\begin{array}{ccc}-1 & -1 & -1 \\ 1 & -3 & 0 \\ 0 & 1 & 1\end{array}\right]$ .

STEP 1

STEP 2

Write down the given matrices $A$ and $B$ .
$A = \left[\begin{array}{ccc} 0 & 0 & 4 \\ -2 & -2 & 2 \\ 0 & -2 & -5 \end{array}\right], \quad B = \left[\begin{array}{ccc} -1 & -1 & -1 \\ 1 & -3 & 0 \\ 0 & 1 & 1 \end{array}\right]$

STEP 3

We will calculate the product $AB$ by computing each element $(AB)_{ij}$ of the resulting matrix.

STEP 4

Calculate the element $(AB)_{11}$ .
$(AB)_{11} = A_{11}B_{11} + A_{12}B_{21} + A_{13}B_{31} = 0 \cdot (-1) + 0 \cdot 1 + 4 \cdot 0 = 0$

STEP 5

Calculate the element $(AB)_{12}$ .
$(AB)_{12} = A_{11}B_{12} + A_{12}B_{22} + A_{13}B_{32} = 0 \cdot (-1) + 0 \cdot (-3) + 4 \cdot 1 = 4$

STEP 6

Calculate the element $(AB)_{13}$ .
$(AB)_{13} = A_{11}B_{13} + A_{12}B_{23} + A_{13}B_{33} = 0 \cdot (-1) + 0 \cdot 0 + 4 \cdot 1 = 4$

STEP 7

Calculate the element $(AB)_{21}$ .
$(AB)_{21} = A_{21}B_{11} + A_{22}B_{21} + A_{23}B_{31} = (-2) \cdot (-1) + (-2) \cdot 1 + 2 \cdot 0 = 2 - 2 + 0 = 0$

STEP 8

Calculate the element $(AB)_{22}$ .
$(AB)_{22} = A_{21}B_{12} + A_{22}B_{22} + A_{23}B_{32} = (-2) \cdot (-1) + (-2) \cdot (-3) + 2 \cdot 1 = 2 + 6 + 2 = 10$

STEP 9

Calculate the element $(AB)_{23}$ .
$(AB)_{23} = A_{21}B_{13} + A_{22}B_{23} + A_{23}B_{33} = (-2) \cdot (-1) + (-2) \cdot 0 + 2 \cdot 1 = 2 + 0 + 2 = 4$

STEP 10

Calculate the element $(AB)_{31}$ .
$(AB)_{31} = A_{31}B_{11} + A_{32}B_{21} + A_{33}B_{31} = 0 \cdot (-1) + (-2) \cdot 1 + (-5) \cdot 0 = 0 - 2 + 0 = -2$

STEP 11

Calculate the element $(AB)_{32}$ .
$(AB)_{32} = A_{31}B_{12} + A_{32}B_{22} + A_{33}B_{32} = 0 \cdot (-1) + (-2) \cdot (-3) + (-5) \cdot 1 = 0 + 6 - 5 = 1$

STEP 12

Calculate the element $(AB)_{33}$ .
$(AB)_{33} = A_{31}B_{13} + A_{32}B_{23} + A_{33}B_{33} = 0 \cdot (-1) + (-2) \cdot 0 + (-5) \cdot 1 = 0 + 0 - 5 = -5$

STEP 13

Combine all the calculated elements to form the resulting matrix $AB$ .
$AB = \left[\begin{array}{ccc} (AB)_{11} & (AB)_{12} & (AB)_{13} \\ (AB)_{21} & (AB)_{22} & (AB)_{23} \\ (AB)_{31} & (AB)_{32} & (AB)_{33} \end{array}\right]$

STEP 14

Substitute the calculated values into the resulting matrix.
$AB = \left[\begin{array}{ccc} 0 & 4 & 4 \\ 0 & 10 & 4 \\ -2 & 1 & -5 \end{array}\right]$
The product of the given matrices is:
$\left[\begin{array}{ccc} 0 & 4 & 4 \\ 0 & 10 & 4 \\ -2 & 1 & -5 \end{array}\right]$

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STEP 1

STEP 2

STEP 3

We will calculate the product ABABAB by computing each element (AB)ij(AB)_{ij}(AB)ij​ of the resulting matrix.

STEP 4

Calculate the element (AB)11(AB)_{11}(AB)11​.(AB)11=A11B11+A12B21+A13B31=0⋅(−1)+0⋅1+4⋅0=0(AB)_{11} = A_{11}B_{11} + A_{12}B_{21} + A_{13}B_{31} = 0 \cdot (-1) + 0 \cdot 1 + 4 \cdot 0 = 0(AB)11​=A11​B11​+A12​B21​+A13​B31​=0⋅(−1)+0⋅1+4⋅0=0

STEP 5

Calculate the element (AB)12(AB)_{12}(AB)12​.(AB)12=A11B12+A12B22+A13B32=0⋅(−1)+0⋅(−3)+4⋅1=4(AB)_{12} = A_{11}B_{12} + A_{12}B_{22} + A_{13}B_{32} = 0 \cdot (-1) + 0 \cdot (-3) + 4 \cdot 1 = 4(AB)12​=A11​B12​+A12​B22​+A13​B32​=0⋅(−1)+0⋅(−3)+4⋅1=4

STEP 6

Calculate the element (AB)13(AB)_{13}(AB)13​.(AB)13=A11B13+A12B23+A13B33=0⋅(−1)+0⋅0+4⋅1=4(AB)_{13} = A_{11}B_{13} + A_{12}B_{23} + A_{13}B_{33} = 0 \cdot (-1) + 0 \cdot 0 + 4 \cdot 1 = 4(AB)13​=A11​B13​+A12​B23​+A13​B33​=0⋅(−1)+0⋅0+4⋅1=4

STEP 7

STEP 8

STEP 9

STEP 10

STEP 11

STEP 12

STEP 13

STEP 14

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We will calculate the product $AB$ by computing each element $(AB)_{ij}$ of the resulting matrix.

Calculate the element $(AB)_{11}$ .
$(AB)_{11} = A_{11}B_{11} + A_{12}B_{21} + A_{13}B_{31} = 0 \cdot (-1) + 0 \cdot 1 + 4 \cdot 0 = 0$

Calculate the element $(AB)_{12}$ .
$(AB)_{12} = A_{11}B_{12} + A_{12}B_{22} + A_{13}B_{32} = 0 \cdot (-1) + 0 \cdot (-3) + 4 \cdot 1 = 4$

Calculate the element $(AB)_{13}$ .
$(AB)_{13} = A_{11}B_{13} + A_{12}B_{23} + A_{13}B_{33} = 0 \cdot (-1) + 0 \cdot 0 + 4 \cdot 1 = 4$