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Precalculus : Find the Product of Two Matrices

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Example Questions

Example Question #572 : Pre Calculus

Calculate AB when

$A=\begin{bmatrix}-1 & 3\\0 & 3\\2 &1 \end{bmatrix}$ and $B=\begin{bmatrix} -2& 4\\ 3&1 \end{bmatrix}$ .

Possible Answers:

$AB=\begin{bmatrix} 9&-5 \\ -3&0 \\ -6&4 \end{bmatrix}$

$AB=\begin{bmatrix} 11& -1\\ 9& 3\\ -1&9 \end{bmatrix}$

$AB=\begin{bmatrix} 3&-2 \\ 5&0 \end{bmatrix}$

$AB=\begin{bmatrix} 6& -7\\ 5& 12\\ -1&3 \end{bmatrix}$

$AB=\begin{bmatrix} 4&3 \\ -2&4 \end{bmatrix}$

Correct answer:

$AB=\begin{bmatrix} 11& -1\\ 9& 3\\ -1&9 \end{bmatrix}$

Explanation:

In order to perform matrix multiplication, the number of columns in the first matrix has to be the same as the number of rows in the second column.

$A:3x2\:\, B:2x2$

$AB=\begin{bmatrix}-1 & 3\\0 & 3\\2 &1 \end{bmatrix}\cdot \begin{bmatrix} -2&4 \\ 3&1 \end{bmatrix}$

From here, we multiply each term in the first matrix's row by the first column in the second matrix. Continue in this fashion to get the product matrix.

$AB=\begin{bmatrix} (-1)(-2)+(3)(3)& (-1)(4)+(3)(1)\\ (0)(-2)+(3)(3)&(0)(4)+(3)(1) \\ (2)(-2)+(1)(3)&(2)(4)+(1)(1) \end{bmatrix}$

$AB=\begin{bmatrix} 11& -1\\ 9& 3\\ -1&9 \end{bmatrix}$

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Example Question #68 : Matrices And Vectors

Evaluate: $\begin{bmatrix} 2&7 \\ 6&-5 \end{bmatrix} \begin{bmatrix} 3& 4\\ 2&1 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 25&30 \\ -3&-4\end{bmatrix}$

$\begin{bmatrix} 20&15 \\ 8&19 \end{bmatrix}$

$\begin{bmatrix}34&11 \\ -2&7 \end{bmatrix}$

$\begin{bmatrix}11&34 \\ 7&-2 \end{bmatrix}$

$\begin{bmatrix} 6&28 \\ 12&-5 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 20&15 \\ 8&19 \end{bmatrix}$

Explanation:

Write the rule for multiplying a two by two matrix. The result will be a two by two matrix.

$\begin{bmatrix} a& b\\ c&d\end{bmatrix} \begin{bmatrix} w&x \\ y&z \end{bmatrix} = \begin{bmatrix} aw+by&ax+bz \\ cw+dy&cx+dz \end{bmatrix}$

Follow this rule for the given problem.

$\begin{bmatrix} 2&7 \\ 6&-5 \end{bmatrix} \begin{bmatrix} 3& 4\\ 2&1 \end{bmatrix}= \begin{bmatrix} (2)(3)+(7)(2)&(2)(4)+(7)(1) \\ (6)(3)+(-5)(2)&(6)(4)+(-5)(1) \end{bmatrix}$

Simplify, and the answer is: $\begin{bmatrix} 20&15 \\ 8&19 \end{bmatrix}$

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Example Question #31 : Multiplication Of Matrices

Multiply the matrices: $\begin{bmatrix} 1& 3 \end{bmatrix}\begin{bmatrix} 4&9 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 5& 12 \end{bmatrix}$

$\textup{Cannot be solved.}$

$\begin{bmatrix} 4& 27 \end{bmatrix}$

108

Correct answer:

$\textup{Cannot be solved.}$

Explanation:

In order to multiply these matrices we will need to consider the rows and columns for each matrix.

Both matrices have a dimension of $1\times 2$ .

The rule for multiplying matrices is where the number of columns of the first matrix must match the number of rows of the second matrix.

If the dimensions of the first matrix are $a\times b$ , and the dimensions of the second matrix are $b\times c$ , then we will get a dimension of $a \times c$ matrix as a result. If the value of are not matched, we cannot evaluate the product of the matrices.

The correct answer is: $\textup{Cannot be solved.}$

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Example Question #31 : Multiplication Of Matrices

Find the product of A and B.

$A=\begin{bmatrix} 3 &-1 &4 \end{bmatrix}$

$B=\begin{bmatrix} 2\\ 1\\-3 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 1 \end{bmatrix}$

$\begin{bmatrix} 6 & -2& -1\\4 &3 &-6 \\2 &5 &6 \end{bmatrix}$

$\begin{bmatrix} -7 \end{bmatrix}$

$\begin{bmatrix} 6 &-1 &-12 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} -7 \end{bmatrix}$

Explanation:

Since you are multiplying a $1\times 3$ to a $3\times 1$ , the answer is going to be a $1\times 1$ .

To solve, simply multiply each corresponding element and add together.

Thus, your answer is

[-7] .

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Precalculus : Find the Product of Two Matrices

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #572 : Pre Calculus

Example Question #68 : Matrices And Vectors

Example Question #31 : Multiplication Of Matrices

Example Question #31 : Multiplication Of Matrices

All Precalculus Resources

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