Multiply Matrices 2
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Question 1 of 4
1. Question
Solve`[[1,2,0],[0,-1,3],[0,1,0]]times[[3],[4],[1]]`Hint
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Multiplying Two Matrices
`[[a,b,c]]xx[[p],[q],[r]]=ap+bq+cr`Two matrices can be multiplied only if the number of columns (`n`) in the first matrix is equal to the number of rows
(`m`) in the second matrix.First, check the dimensions of each matrixMatrix `1`:`[[1,2,0],[0,-1,3],[0,1,0]]`rows(`m`)`=3`columns(`n`)`=3`Dimension`(mtimesn)=3xx``3`Matrix `2`:`[[3],[4],[1]]`rows(`m`)`=3`columns(`n`)`=1`Dimension`(mtimesn)=``3``xx1`Since the number of columns in the first matrix (`3`) and the number of rows in the second matrix
(`3`) are equal, these two matrices can be multipliedNext, proceed with multiplying the two matrices`[[1,2,0],[0,-1,3],[0,1,0]]times``[[3],[4],[1]]` `=` \begin{bmatrix}
(1\cdot\color{#9a00c7}{3})+(2\cdot\color{#9a00c7}{4})+(0\cdot\color{#9a00c7}{1}) \\[0.3em]
(0\cdot\color{#9a00c7}{3})+(-1\cdot\color{#9a00c7}{4})+(3\cdot\color{#9a00c7}{1}) \\[0.3em]
(0\cdot\color{#9a00c7}{3})+(1\cdot\color{#9a00c7}{4})+(0\cdot\color{#9a00c7}{1})
\end{bmatrix}`=` `[[3+8+0],[0+(-4)+3],[0+4+0]]` `=` `[[11],[-1],[4]]` `[[11],[-1],[4]]` -
Question 2 of 4
2. Question
Solve`[[4,2,1],[3,-5,2]]times[[3,2,4],[1,4,5],[2,6,1]]`Hint
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Correct!
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Multiplying Two Matrices
`[[a,b,c]]xx[[p],[q],[r]]=ap+bq+cr`Two matrices can be multiplied only if the number of columns (`n`) in the first matrix is equal to the number of rows
(`m`) in the second matrix.First, check the dimensions of each matrixMatrix `1`:`[[4,2,1],[3,-5,2]]`rows(`m`)`=2`columns(`n`)`=3`Dimension`(mtimesn)=2xx``3`Matrix `2`:`[[3,2,4],[1,4,5],[2,6,1]]`rows(`m`)`=3`columns(`n`)`=3`Dimension`(mtimesn)=``3``xx3`Since the number of columns in the first matrix (`3`) and the number of rows in the second matrix
(`3`) are equal, these two matrices can be multipliedNext, proceed with multiplying the two matrices`[[4,2,1],[3,-5,2]]times``[[3,2,4],[1,4,5],[2,6,1]]` `=` \begin{bmatrix}
(4\cdot\color{#9a00c7}{3})+(2\cdot\color{#9a00c7}{1})+(1\cdot\color{#9a00c7}{2}) & (4\cdot\color{#9a00c7}{2})+(2\cdot\color{#9a00c7}{4})+(1\cdot\color{#9a00c7}{6}) & (4\cdot\color{#9a00c7}{4})+(2\cdot\color{#9a00c7}{5})+(1\cdot\color{#9a00c7}{1}) \\[0.3em]
(3\cdot\color{#9a00c7}{3})+(-5\cdot\color{#9a00c7}{1})+(2\cdot\color{#9a00c7}{2}) & (3\cdot\color{#9a00c7}{2})+(-5\cdot\color{#9a00c7}{4})+(2\cdot\color{#9a00c7}{6}) & (3\cdot\color{#9a00c7}{4})+(-5\cdot\color{#9a00c7}{5})+(2\cdot\color{#9a00c7}{1})
\end{bmatrix}`=` `[[12+2+2,8+8+6,16+10+1],[9+(-5)+4,6+(-20)+12,12+(-25)+2]]` `=` `[[16,22,27],[8,-2,-11]]` `[[16,22,27],[8,-2,-11]]` -
Question 3 of 4
3. Question
Solve for `2xy`, given that:`x=[[3,-1],[5,-2]]` `y=[[1,0],[-4,3]]`Hint
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Incorrect
Multiplying Two Matrices
`[[a,b,c]]xx[[p],[q],[r]]=ap+bq+cr`Two matrices can be multiplied only if the number of columns (`n`) in the first matrix is equal to the number of rows
(`m`) in the second matrix.First, check the dimensions of matrices `x` and `y`Matrix `x`:`[[3,-1],[5,-2]]`rows(`m`)`=2`columns(`n`)`=2`Dimension`(mtimesn)=2xx``2`Matrix `y`:`[[1,0],[-4,3]]`rows(`m`)`=2`columns(`n`)`=2`Dimension`(mtimesn)=``2``xx2`Since the number of columns in the first matrix (`2`) and the number of rows in the second matrix
(`2`) are equal, these two matrices can be multipliedNext, substitute the two matrices to `2xy` and solve`2xy` `=` `2times[[3,-1],[5,-2]]times``[[1,0],[-4,3]]` `=` `[[6,-2],[10,-4]]times``[[1,0],[-4,3]]` `=` \begin{bmatrix}
(6\cdot\color{#9a00c7}{1})+(-2\cdot\color{#9a00c7}{-4}) & (6\cdot\color{#9a00c7}{0})+(-2\cdot\color{#9a00c7}{3}) \\[0.3em]
(10\cdot\color{#9a00c7}{1})+(-4\cdot\color{#9a00c7}{-4}) & (10\cdot\color{#9a00c7}{0})+(-4\cdot\color{#9a00c7}{3})
\end{bmatrix}`=` `[[6+8,0+(-6)],[10+16,0+(-12)]]` `=` `[[14,-6],[26,-12]]` `[[14,-6],[26,-12]]` -
Question 4 of 4
4. Question
Solve for `x(x+y)`, given that:`x=[[3,-1],[5,-2]]` `y=[[1,0],[-4,3]]`Hint
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Excellent!
Incorrect
Multiplying Two Matrices
`[[a,b,c]]xx[[p],[q],[r]]=ap+bq+cr`Two matrices can be multiplied only if the number of columns (`n`) in the first matrix is equal to the number of rows
(`m`) in the second matrix.Two matrices can be added only if their dimensions (`mtimesn`) are equal.First, check the dimensions of matrices `x` and `y`Matrix `x`:`[[3,-1],[5,-2]]`rows(`m`)`=2`columns(`n`)`=2`Dimension`(mtimesn)=2xx``2`Matrix `y`:`[[1,0],[-4,3]]`rows(`m`)`=2`columns(`n`)`=2`Dimension`(mtimesn)=``2``xx2`Since the number of columns in the first matrix (`2`) and the number of rows in the second matrix
(`2`) are equal, these two matrices can be multipliedAlso note that since their dimensions are equal, they can be addedNext, substitute the two matrices to `x(x+y)` and solve`x(x+y)` `=` `[[3,-1],[5,-2]]times([[3,-1],[5,-2]]+[[1,0],[-4,3]])` `=` `[[3,-1],[5,-2]]times[[3+1,-1+0],[5+(-4),-2+3]]` `=` `[[3,-1],[5,-2]]times``[[4,-1],[1,1]]` `=` \begin{bmatrix}
(3\cdot\color{#9a00c7}{4})+(-1\cdot\color{#9a00c7}{1}) & (3\cdot\color{#9a00c7}{-1})+(-1\cdot\color{#9a00c7}{1}) \\[0.3em]
(5\cdot\color{#9a00c7}{4})+(-2\cdot\color{#9a00c7}{1}) & (5\cdot\color{#9a00c7}{-1})+(-2\cdot\color{#9a00c7}{1})
\end{bmatrix}`=` `[[12+(-1),-3+(-1)],[20+(-2),-5+(-2)]]` `=` `[[11,-4],[18,-7]]` `[[11,-4],[18,-7]]`
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- Multiply Matrices 1
- Multiply Matrices 2
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- Determinant of a Matrix
- Inverse of a Matrix
- Matrices: Systems of Equations 1
- Matrices: Systems of Equations 2
- Gauss Jordan Elimination
- Cramer’s Rule