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Linear Algebra : Linear Algebra

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Example Questions

Example Question #32 : Matrices

If is a $3 \times 2$ matrix and is a $2 \times 4$ matrix, can the product $A \times B$ be multiplied? What about $B \times A$ ?

Possible Answers:

$A \times B$ can be multiplied

$B \times A$ cannot be multiplied

$A \times B$ can be multiplied

$B \times A$ can be multiplied

$A \times B$ cannot be multiplied

$B \times A$ cannot be multiplied

$A \times B$ cannot be multiplied

$B \times A$ can be multiplied

Correct answer:

$A \times B$ can be multiplied

$B \times A$ cannot be multiplied

Explanation:

If is an $(m\times n)$ matrix and is an $(r\times s)$ matrix, $(A\times B)$ can only be multiplied if n=r .

Since is a $3 \times 2$ matrix and is a $2 \times 4$ matrix, n=r and $A \times B$ can be multiplied.

has rows and has columns, therefore $B \times A$ cannot be multiplied.

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Example Question #32 : Matrices

If is a $3 \times 4$ matrix and is a $4 \times 4$ matrix, what are the dimensions of the product $A \times B$ ?

Possible Answers:

$3 \times 3$

$4 \times 4$

$3 \times 4$

$A \times B$ cannot be multiplied

Correct answer:

$3 \times 4$

Explanation:

If is an $(m\times n)$ matrix and is an $(n\times s)$ matrix, the dimensions of $(A\times B)$ are $(m\times s)$ .

In this problem, If is a $3 \times 4$ matrix and is a $4 \times 4$ matrix, so

the dimensions of $(A\times B)$ are $(3\times 4)$ .

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Example Question #31 : Matrix Matrix Product

If is a $2 \times 5$ matrix and is a $5 \times 2$ matrix, what are the dimensions of the product $A \times B$ ?

Possible Answers:

$(5\times 2)$

$(2\times 2)$

$A \times B$ cannot be multiplied

$(2\times 5)$

Correct answer:

$(2\times 2)$

Explanation:

If is an $(m\times n)$ matrix and is an $(n\times s)$ matrix, the dimensions of $(A\times B)$ are $(m\times s)$ .

In this problem, If is a $2 \times 5$ matrix and is a $5 \times 2$ matrix, so

the dimensions of $(A\times B)$ are $(2\times 2)$ .

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Example Question #34 : Matrices

Find the product $A \times B$ .

$A=\begin{bmatrix} -1 &5\\ 4 &-4 \end{bmatrix}$ , $B=\begin{bmatrix} 2 &-1&5\\ 0 &-7&4 \end{bmatrix}$

Possible Answers:

$A \times B=\begin{bmatrix} -2 &-5&5\\ 0 &28&4 \end{bmatrix}$

$A \times B=\begin{bmatrix} -2& -34& 15\\ 8& 24&4 \end{bmatrix}$

$A \times B$ cannot be multiplied

$A \times B=\begin{bmatrix} -1 &5&5\\ 4&-4&4 \end{bmatrix}$

Correct answer:

$A \times B=\begin{bmatrix} -2& -34& 15\\ 8& 24&4 \end{bmatrix}$

Explanation:

If is an $(m\times n)$ matrix and is an $(r\times s)$ matrix,

$(A\times B)$ can only be multiplied if n=r , or the number of columns in equals the number of rows in . Otherwise there is a mismatch, and the two matrices can not be multiplied. $(A\times B)$ will be an $(m\times s)$ matrix

Since is a $2\times2$ matrix and is a $2\times3$ matrix, then $(A\times B)$ can be multiplied and will have the dimensions $2\times3$ .

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}& ab_{13}\\ ab_{21}& ab_{22}& ab_{23} \end{bmatrix}$

To find the product $A \times B$ , you must find the dot product of the rows of and the columns of

$A=\begin{bmatrix} -1 &5\\ 4 &-4 \end{bmatrix}$ , $B=\begin{bmatrix} 2 &-1&5\\ 0 &-7&4 \end{bmatrix}$

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}& ab_{13}\\ ab_{21}& ab_{22}& ab_{23} \end{bmatrix}$

We find $ab_{11}$ by finding the dot product of the row of and column of .

$ab_{11}=(-1)(2)+(5)(0)=-2+0=-2$

We find $ab_{12}$ by finding the dot product of the row of and column of .

$ab_{12}=(-1)(-1)+(5)(-7)=1-35=-34$

We use the same method to find the rest of the matrix values

$ab_{13}=(-1)(5)+(5)(4)=-5+20=15$

$ab_{21}=(4)(2)+(-4)(0)=8+0=8$

$ab_{22}=(4)(-1)+(-4)(-7)=-4+28=24$

$ab_{23}=(4)(5)+(-4)(4)=20-16=4$

$A \times B=\begin{bmatrix} -2& -34& 15\\ 8& 24&4 \end{bmatrix}$

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Example Question #35 : Matrix Matrix Product

Find the product $A \times B$ .

$A=\begin{bmatrix} -1& 2\\ 4& 3 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1\\ -2& 5 \end{bmatrix}$

Possible Answers:

$A \times B=\begin{bmatrix} -7& -2\\ -8& 15 \end{bmatrix}$

$A \times B=\begin{bmatrix} -11& 11\\ 22& 11 \end{bmatrix}$

$A\times B$ cannot be multiplied

$A \times B=\begin{bmatrix} 6& 1\\ 2& 8 \end{bmatrix}$

Correct answer:

$A \times B=\begin{bmatrix} -11& 11\\ 22& 11 \end{bmatrix}$

Explanation:

If is an $(m\times n)$ matrix and is an $(r\times s)$ matrix,

Since is a $2\times2$ matrix and is a $2\times2$ matrix, then $(A\times B)$ can be multiplied and will have the dimensions $2\times2$ .

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}\\ ab_{21}& ab_{22} \end{bmatrix}$

To find the product $A \times B$ , you must find the dot product of the rows of and the columns of

$A=\begin{bmatrix} -1& 2\\ 4& 3 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1\\ -2& 5 \end{bmatrix}$

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}\\ ab_{21}& ab_{22} \end{bmatrix}$

We find $ab_{11}$ by finding the dot product of the row of and column of .

$ab_{11}=(-1)(7)+(2)(-2)=-7-4=-11$

We find $ab_{12}$ by finding the dot product of the row of and column of .

$ab_{12}=(-1)(-1)+(2)(5)=1+10=11$

We use the same method to find the rest of the matrix values

$ab_{21}=(4)(7)+(3)(-2)=28-6=22$

$ab_{22}=(4)(-1)+(3)(5)=-4+15=11$

$A \times B=\begin{bmatrix} -11& 11\\ 22& 11 \end{bmatrix}$

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Example Question #31 : Matrix Matrix Product

Find the product $A \times B$ .

$A=\begin{bmatrix} -1& 2&0\\ 4& 3&6 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1\\ -2& 5 \\ 1 & 3 \end{bmatrix}$

Possible Answers:

$A \times B=\begin{bmatrix} 2& 10\\ 4& 9 \end{bmatrix}$

$A \times B=\begin{bmatrix} -1& 2\\ -2& 5 \end{bmatrix}$

$A\times B$ cannot be multiplied

$A \times B=\begin{bmatrix} -11& 11\\ 28& 29 \end{bmatrix}$

Correct answer:

$A \times B=\begin{bmatrix} -11& 11\\ 28& 29 \end{bmatrix}$

Explanation:

If is an $(m\times n)$ matrix and is an $(r\times s)$ matrix,

Since is a $2\times3$ matrix and is a $3\times2$ matrix, then $(A\times B)$ can be multiplied and will have the dimensions $2\times2$ .

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}\\ ab_{21}& ab_{22} \end{bmatrix}$

To find the product $A \times B$ , you must find the dot product of the rows of and the columns of

$A=\begin{bmatrix} -1& 2&0\\ 4& 3&6 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1\\ -2& 5 \\ 1 & 3 \end{bmatrix}$

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}\\ ab_{21}& ab_{22} \end{bmatrix}$

We find $ab_{11}$ by finding the dot product of the row of and column of .

$ab_{11}=(-1)(7)+(2)(-2)+(0)(1)=-7-4+0=-11$

We find $ab_{12}$ by finding the dot product of the row of and column of .

$ab_{12}=(-1)(-1)+(2)(5)+(0)(3)=1+10+0=11$

We use the same method to find the rest of the matrix values

$ab_{21}=(4)(7)+(3)(-2)+(6)(1)=28-6+6=28$

$ab_{22}=(4)(-1)+(3)(5)+(6)(3)=-4+15+18=29$

$A \times B=\begin{bmatrix} -11& 11\\ 28& 29 \end{bmatrix}$

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Example Question #41 : Matrix Matrix Product

Find the product $A \times B$ .

$A=\begin{bmatrix} -1& 2\\ 4& 3 \\ 5& -1 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1\\ -2& 5 \end{bmatrix}$

Possible Answers:

$A \times B=\begin{bmatrix} 7& -1\\ -2& 5 \\ 5& -1 \end{bmatrix}$

$A \times B=\begin{bmatrix} -7& -2\\ -8& 15 \\ 5& -1 \end{bmatrix}$

$A \times B$ cannot be multiplied

$A \times B=\begin{bmatrix} -11& 11\\ 22& 11\\ 37& 10 \end{bmatrix}$

Correct answer:

$A \times B=\begin{bmatrix} -11& 11\\ 22& 11\\ 37& 10 \end{bmatrix}$

Explanation:

If is an $(m\times n)$ matrix and is an $(r\times s)$ matrix,

Since is a $3\times2$ matrix and is a $2\times2$ matrix, then $(A\times B)$ can be multiplied and will have the dimensions $3\times2$ .

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}\\ ab_{21}& ab_{22}\\ ab_{31}& ab_{32} \end{bmatrix}$

To find the product $A \times B$ , you must find the dot product of the rows of and the columns of

$A=\begin{bmatrix} -1& 2\\ 4& 3 \\ 5& -1 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1\\ -2& 5 \end{bmatrix}$

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}\\ ab_{21}& ab_{22}\\ ab_{31}& ab_{32} \end{bmatrix}$

We find $ab_{11}$ by finding the dot product of the row of and column of .

$ab_{11}=(-1)(7)+(2)(-2)=-7-4=-11$

We find $ab_{12}$ by finding the dot product of the row of and column of .

$ab_{12}=(-1)(-1)+(2)(5)=1+10=11$

We use the same method to find the rest of the matrix values

$ab_{21}=(4)(7)+(3)(-2)=28-6=22$

$ab_{22}=(4)(-1)+(3)(5)=-4+15=11$

$ab_{31}=(5)(7)+(-1)(-2)=35+2=37$

$ab_{32}=(5)(-1)+(-1)(5)=-5-5=10$

$A \times B=\begin{bmatrix} -11& 11\\ 22& 11\\ 37& 10 \end{bmatrix}$

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Example Question #42 : Matrices

Find the product $A \times B$ .

$A=\begin{bmatrix} -1& 2 &5\\ 4& 3 &1 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1\\ -2& 5 \end{bmatrix}$

Possible Answers:

$A \times B=\begin{bmatrix} -7& -2 \\ -8& 15 \end{bmatrix}$

$A \times B=\begin{bmatrix} 6& 1 &5\\ 2& 8 &1 \end{bmatrix}$

$A \times B$ cannot be multiplied.

$A \times B=\begin{bmatrix} -7& -2 &5\\ -8& 15 &1 \end{bmatrix}$

Correct answer:

$A \times B$ cannot be multiplied.

Explanation:

If is an $(m\times n)$ matrix and is an $(r\times s)$ matrix,

Since is a $2\times3$ matrix and is a $2\times2$ matrix, $n\neq r$ so $(A\times B)$ cannot be multiplied.

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Example Question #723 : Linear Algebra

Find the product $A \times B$ .

$A=\begin{bmatrix} -1 &5\\ 4 &1 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1& 0\\ -2& 5&8 \end{bmatrix}$

Possible Answers:

$A \times B=\begin{bmatrix} -7& -5& 0\\ -8& 5&8 \end{bmatrix}$

$A \times B=\begin{bmatrix} -1& 5& 0\\ 4& 1&8 \end{bmatrix}$

$A \times B$ cannot be multiplied

$A \times B=\begin{bmatrix} -17& 26& 40\\ 26& 1&8 \end{bmatrix}$

Correct answer:

$A \times B=\begin{bmatrix} -17& 26& 40\\ 26& 1&8 \end{bmatrix}$

Explanation:

If is an $(m\times n)$ matrix and is an $(r\times s)$ matrix,

Since is a $2\times2$ matrix and is a $2\times3$ matrix, then $(A\times B)$ can be multiplied and will have the dimensions $2\times3$ .

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}& ab_{13}\\ ab_{21}& ab_{22}& ab_{23} \end{bmatrix}$

To find the product $A \times B$ , you must find the dot product of the rows of and the columns of

$A=\begin{bmatrix} -1 &5\\ 4 &1 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1& 0\\ -2& 5&8 \end{bmatrix}$

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}& ab_{13}\\ ab_{21}& ab_{22}& ab_{23} \end{bmatrix}$

We find $ab_{11}$ by finding the dot product of the row of and column of .

$ab_{11}=(-1)(7)+(5)(-2)=-7-10=-17$

We find $ab_{12}$ by finding the dot product of the row of and column of .

$ab_{12}=(-1)(-1)+(5)(5)=1+25=26$

We use the same method to find the rest of the matrix values

$ab_{13}=(-1)(0)+(5)(8)=0+40=40$

$ab_{21}=(4)(7)+(1)(-2)=28-2=26$

$ab_{22}=(4)(-1)+(1)(5)=-4+5=1$

$ab_{23}=(4)(0)+(1)(8)=0+8=8$

$A \times B=\begin{bmatrix} -17& 26& 40\\ 26& 1&8 \end{bmatrix}$

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Example Question #724 : Linear Algebra

Find the product $A \times B$ .

$A=\begin{bmatrix} -1 &5&2\\ 4 &-4&5 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1& 0\\ -2& 5&8\\ 1& 2&4 \end{bmatrix}$

Possible Answers:

$A \times B=\begin{bmatrix} -15& 30& 48\\ 41& -14&-12 \end{bmatrix}$

$A \times B=\begin{bmatrix} -7 &-5&0\\ -8 &-20&40 \end{bmatrix}$

$A \times B$ cannot be multiplied

$A \times B=\begin{bmatrix} 7& -1& 0\\ -2& 5&8 \end{bmatrix}$

Correct answer:

$A \times B=\begin{bmatrix} -15& 30& 48\\ 41& -14&-12 \end{bmatrix}$

Explanation:

If is an $(m\times n)$ matrix and is an $(r\times s)$ matrix,

Since is a $2\times3$ matrix and is a $3\times3$ matrix, then $(A\times B)$ can be multiplied and will have the dimensions $2\times3$ .

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}& ab_{13}\\ ab_{21}& ab_{22}& ab_{23} \end{bmatrix}$

To find the product $A \times B$ , you must find the dot product of the rows of and the columns of

$A=\begin{bmatrix} -1 &5&2\\ 4 &-4&5 \end{bmatrix}$ , $B=\begin{bmatrix} 7& -1& 0\\ -2& 5&8\\ 1& 2&4 \end{bmatrix}$

$A \times B=\begin{bmatrix} ab_{11}& ab_{12}& ab_{13}\\ ab_{21}& ab_{22}& ab_{23} \end{bmatrix}$

We find $ab_{11}$ by finding the dot product of the row of and column of .

$ab_{11}=(-1)(7)+(5)(-2)+(2)(1)=-7-10+2=-15$

We find $ab_{12}$ by finding the dot product of the row of and column of .

$ab_{12}=(-1)(-1)+(5)(5)+(2)(2)=1+25+4=30$

We use the same method to find the rest of the matrix values

$ab_{13}=(-1)(0)+(5)(8)+(2)(4)=0+40+8=48$

$ab_{21}=(4)(7)+(-4)(-2)+(5)(1)=28+8+5=41$

$ab_{22}=(4)(-1)+(-4)(5)+(5)(2)=-4-20+10=-14$

$ab_{23}=(4)(0)+(-4)(8)+(5)(4)=0-32+20=-12$

$A \times B=\begin{bmatrix} -15& 30& 48\\ 41& -14&-12 \end{bmatrix}$

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Linear Algebra : Linear Algebra

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #32 : Matrices

Example Question #32 : Matrices

Example Question #31 : Matrix Matrix Product

Example Question #34 : Matrices

Example Question #35 : Matrix Matrix Product

Example Question #31 : Matrix Matrix Product

Example Question #41 : Matrix Matrix Product

Example Question #42 : Matrices

Example Question #723 : Linear Algebra

Example Question #724 : Linear Algebra

All Linear Algebra Resources

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