Solved on Oct 31, 2023

Find the product of two $2 \times 2$ matrices $X$ and $Y$ , where $X=\left[\begin{array}{cc}4 & 4 \\ -3 & -1\end{array}\right]$ and $Y=\left[\begin{array}{cc}4 & 1 \\ 4 & -1\end{array}\right]$ .

STEP 1

Assumptions1. The matrices X and Y are bothx matrices. . The multiplication of two matrices is defined as the dot product of the rows of the first matrix with the columns of the second matrix.

STEP 2

First, we need to define the multiplication operation for two matrices. If A is an n×m matrix and B is an m×p matrix, then their matrix product AB is an n×p matrix.
In this case, both X and Y are2x2 matrices, so their product will also be a2x2 matrix.

STEP 3

The elements of the resulting matrix are calculated as follows $(XY)_{ij} = \sum_{k=1}^{2} X_{ik}Y_{kj}$ where i and j are the row and column indices, respectively.

STEP 4

Let's calculate the elements of the resulting matrix. Start with the element at the first row and first column ((XY)_{11}) $(XY)_{11} = X_{11}Y_{11} + X_{12}Y_{21}$

STEP 5

Plug in the values from matrices X and Y $(XY)_{11} =44 +44$

STEP 6

Calculate the result $(XY)_{11} =16 +16 =32$

STEP 7

Now, calculate the element at the first row and second column ((XY)_{12}) $(XY)_{12} = X_{11}Y_{12} + X_{12}Y_{22}$

STEP 8

Plug in the values from matrices X and Y $(XY)_{12} =41 +4(-1)$

STEP 9

Calculate the result $(XY)_{12} =4 -4 =$

STEP 10

Next, calculate the element at the second row and first column ((XY)_{21}) $(XY)_{21} = X_{21}Y_{} + X_{22}Y_{21}$

STEP 11

Plug in the values from matrices X and Y $(XY)_{21} = -34 + -4$

STEP 12

Calculate the result $(XY)_{21} = -12 -4 = -16$

STEP 13

Finally, calculate the element at the second row and second column ((XY)_{22}) $(XY)_{22} = X_{21}Y_{12} + X_{22}Y_{22}$

STEP 14

Plug in the values from matrices X and Y $(XY)_{22} = -3* + -*(-)$

STEP 15

Calculate the result $(XY)_{22} = -3 + = -2$

STEP 16

Now that we have all the elements of the resulting matrix, we can write down the matrix XY $XY = \left[\begin{array}{cc}32 &0 \\ -16 & -2\end{array}\right]$ So, the product of matrices X and Y is the2x2 matrix with elements32,0, -16, and -2.

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Find the product of two $2 \times 2$ matrices $X$ and $Y$ , where $X=\left[\begin{array}{cc}4 & 4 \\ -3 & -1\end{array}\right]$ and $Y=\left[\begin{array}{cc}4 & 1 \\ 4 & -1\end{array}\right]$ .

STEP 1

Assumptions1. The matrices X and Y are bothx matrices. . The multiplication of two matrices is defined as the dot product of the rows of the first matrix with the columns of the second matrix.

STEP 2

First, we need to define the multiplication operation for two matrices. If A is an n×m matrix and B is an m×p matrix, then their matrix product AB is an n×p matrix.
In this case, both X and Y are2x2 matrices, so their product will also be a2x2 matrix.

STEP 3

The elements of the resulting matrix are calculated as follows $(XY)_{ij} = \sum_{k=1}^{2} X_{ik}Y_{kj}$ where i and j are the row and column indices, respectively.

STEP 4

Let's calculate the elements of the resulting matrix. Start with the element at the first row and first column ((XY)_{11}) $(XY)_{11} = X_{11}Y_{11} + X_{12}Y_{21}$

STEP 5

Plug in the values from matrices X and Y $(XY)_{11} =44 +44$

STEP 6

Calculate the result $(XY)_{11} =16 +16 =32$

STEP 7

Now, calculate the element at the first row and second column ((XY)_{12}) $(XY)_{12} = X_{11}Y_{12} + X_{12}Y_{22}$

STEP 8

Plug in the values from matrices X and Y $(XY)_{12} =41 +4(-1)$

STEP 9

Calculate the result $(XY)_{12} =4 -4 =$

STEP 10

Next, calculate the element at the second row and first column ((XY)_{21}) $(XY)_{21} = X_{21}Y_{} + X_{22}Y_{21}$

STEP 11

Plug in the values from matrices X and Y $(XY)_{21} = -34 + -4$

STEP 12

Calculate the result $(XY)_{21} = -12 -4 = -16$

STEP 13

Finally, calculate the element at the second row and second column ((XY)_{22}) $(XY)_{22} = X_{21}Y_{12} + X_{22}Y_{22}$

STEP 14

Plug in the values from matrices X and Y $(XY)_{22} = -3* + -*(-)$

STEP 15

Calculate the result $(XY)_{22} = -3 + = -2$

STEP 16

Now that we have all the elements of the resulting matrix, we can write down the matrix XY $XY = \left[\begin{array}{cc}32 &0 \\ -16 & -2\end{array}\right]$ So, the product of matrices X and Y is the2x2 matrix with elements32,0, -16, and -2.

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Find the product of two 2×22 \times 22×2 matrices XXX and YYY, where X=[44−3−1]X=\left[\begin{array}{cc}4 & 4 \\ -3 & -1\end{array}\right]X=[4−3​4−1​] and Y=[414−1]Y=\left[\begin{array}{cc}4 & 1 \\ 4 & -1\end{array}\right]Y=[44​1−1​].

STEP 1

Assumptions1. The matrices X and Y are bothx matrices. . The multiplication of two matrices is defined as the dot product of the rows of the first matrix with the columns of the second matrix.

STEP 2

First, we need to define the multiplication operation for two matrices. If A is an n×m matrix and B is an m×p matrix, then their matrix product AB is an n×p matrix.In this case, both X and Y are2x2 matrices, so their product will also be a2x2 matrix.

STEP 3

The elements of the resulting matrix are calculated as follows(XY)ij=∑k=12XikYkj (XY)_{ij} = \sum_{k=1}^{2} X_{ik}Y_{kj} (XY)ij​=k=1∑2​Xik​Ykj​where i and j are the row and column indices, respectively.

STEP 4

Let's calculate the elements of the resulting matrix. Start with the element at the first row and first column ((XY)_{11})(XY)11=X11Y11+X12Y21 (XY)_{11} = X_{11}Y_{11} + X_{12}Y_{21} (XY)11​=X11​Y11​+X12​Y21​

STEP 5

Plug in the values from matrices X and Y(XY)11=4∗4+4∗4 (XY)_{11} =4*4 +4*4 (XY)11​=4∗4+4∗4

STEP 6

Calculate the result(XY)11=16+16=32 (XY)_{11} =16 +16 =32 (XY)11​=16+16=32

STEP 7

Now, calculate the element at the first row and second column ((XY)_{12})(XY)12=X11Y12+X12Y22 (XY)_{12} = X_{11}Y_{12} + X_{12}Y_{22} (XY)12​=X11​Y12​+X12​Y22​

STEP 8

Plug in the values from matrices X and Y(XY)12=4∗1+4∗(−1) (XY)_{12} =4*1 +4*(-1) (XY)12​=4∗1+4∗(−1)

STEP 9

Calculate the result(XY)12=4−4= (XY)_{12} =4 -4 = (XY)12​=4−4=

STEP 10

Next, calculate the element at the second row and first column ((XY)_{21})(XY)21=X21Y+X22Y21 (XY)_{21} = X_{21}Y_{} + X_{22}Y_{21} (XY)21​=X21​Y​+X22​Y21​

STEP 11

Plug in the values from matrices X and Y(XY)21=−3∗4+−∗4 (XY)_{21} = -3*4 + -*4 (XY)21​=−3∗4+−∗4

STEP 12

Calculate the result(XY)21=−12−4=−16 (XY)_{21} = -12 -4 = -16 (XY)21​=−12−4=−16

STEP 13

Finally, calculate the element at the second row and second column ((XY)_{22})(XY)22=X21Y12+X22Y22 (XY)_{22} = X_{21}Y_{12} + X_{22}Y_{22} (XY)22​=X21​Y12​+X22​Y22​

STEP 14

Plug in the values from matrices X and Y(XY)22=−3∗+−∗(−) (XY)_{22} = -3* + -*(-) (XY)22​=−3∗+−∗(−)

STEP 15

Calculate the result(XY)22=−3+=−2 (XY)_{22} = -3 + = -2 (XY)22​=−3+=−2

STEP 16

Now that we have all the elements of the resulting matrix, we can write down the matrix XYXY=[320−16−2] XY = \left[\begin{array}{cc}32 &0 \\ -16 & -2\end{array}\right] XY=[32−16​0−2​]So, the product of matrices X and Y is the2x2 matrix with elements32,0, -16, and -2.

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Find the product of two $2 \times 2$ matrices $X$ and $Y$ , where $X=\left[\begin{array}{cc}4 & 4 \\ -3 & -1\end{array}\right]$ and $Y=\left[\begin{array}{cc}4 & 1 \\ 4 & -1\end{array}\right]$ .

First, we need to define the multiplication operation for two matrices. If A is an n×m matrix and B is an m×p matrix, then their matrix product AB is an n×p matrix.
In this case, both X and Y are2x2 matrices, so their product will also be a2x2 matrix.

The elements of the resulting matrix are calculated as follows $(XY)_{ij} = \sum_{k=1}^{2} X_{ik}Y_{kj}$ where i and j are the row and column indices, respectively.

Let's calculate the elements of the resulting matrix. Start with the element at the first row and first column ((XY)_{11}) $(XY)_{11} = X_{11}Y_{11} + X_{12}Y_{21}$

Plug in the values from matrices X and Y $(XY)_{11} =44 +44$

Calculate the result $(XY)_{11} =16 +16 =32$

Now, calculate the element at the first row and second column ((XY)_{12}) $(XY)_{12} = X_{11}Y_{12} + X_{12}Y_{22}$

Plug in the values from matrices X and Y $(XY)_{12} =41 +4(-1)$

Calculate the result $(XY)_{12} =4 -4 =$

Next, calculate the element at the second row and first column ((XY)_{21}) $(XY)_{21} = X_{21}Y_{} + X_{22}Y_{21}$

Plug in the values from matrices X and Y $(XY)_{21} = -34 + -4$

Calculate the result $(XY)_{21} = -12 -4 = -16$

Finally, calculate the element at the second row and second column ((XY)_{22}) $(XY)_{22} = X_{21}Y_{12} + X_{22}Y_{22}$

Plug in the values from matrices X and Y $(XY)_{22} = -3* + -*(-)$

Calculate the result $(XY)_{22} = -3 + = -2$

Now that we have all the elements of the resulting matrix, we can write down the matrix XY $XY = \left[\begin{array}{cc}32 &0 \\ -16 & -2\end{array}\right]$ So, the product of matrices X and Y is the2x2 matrix with elements32,0, -16, and -2.