# Compute the product. begin{bmatrix}0 & -4 & 5 end{bmatrix}begin{bmatrix}x y z end{bmatrix}

Question
Matrices
Compute the product.
$$\begin{bmatrix}0 & -4 & 5 \end{bmatrix}\begin{bmatrix}x \\ y \\ z \end{bmatrix}$$

2021-03-02
Step 1
Matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
In order to multiply two matrices, we need to do dot product of rows to column.
Step 2
We have the matrices as
$$\begin{bmatrix}0 & -4 & 5 \end{bmatrix}\begin{bmatrix}x \\ y \\ z \end{bmatrix}$$
Here number of column for first matrix is same as number of rows of second matrix. Therefore multiplication of these two matrices took place as
$$\begin{bmatrix}0 & -4 & 5 \end{bmatrix}\begin{bmatrix}x \\ y \\ z \end{bmatrix} =\begin{bmatrix}(0)(x)+(-4)(y)+(5)(z) \end{bmatrix}$$
$$\begin{bmatrix} 0-4y+5z \end{bmatrix}$$
$$\begin{bmatrix} -4y+5z \end{bmatrix}$$
Hence, matrix after multiplication of matrices $$\begin{bmatrix}0 & -4 & 5 \end{bmatrix}\begin{bmatrix}x \\ y \\ z \end{bmatrix} \text{ is } \begin{bmatrix} -4y+5z \end{bmatrix}$$

### Relevant Questions

Compute the product AB by the definition of the product of​ matrices, where $$Ab_1 \text{ and } Ab_2$$ are computed​ separately, and by the​ row-column rule for computing AB.
$$A=\begin{bmatrix}-1 & 2 \\2 & 5\\5&-3 \end{bmatrix} , B=\begin{bmatrix}4 & -1 \\-2 & 4 \end{bmatrix}$$
Determine the product AB
AB=?
Write the matrix equation as a system of linear equations without matrices.
$$\begin{bmatrix}-1 & 0&1 \\0 & -1&0\\0&1&1 \end{bmatrix}\begin{bmatrix}x \\ y \\z \end{bmatrix}=\begin{bmatrix}-4 \\ 2\\4 \end{bmatrix}$$
Write the given matrix equation as a system of linear equations without matrices.
$$\begin{bmatrix}-1 & 0&1 \\ 0 & -1 &0 \\ 0&1&1 \end{bmatrix}\begin{bmatrix}x \\ y \\ z \end{bmatrix}=\begin{bmatrix}-4 \\ 2 \\ 4 \end{bmatrix}$$
Compute the indicated matrices, if possible .
A^2B
let $$A=\begin{bmatrix}1 & 2 \\3 & 5 \end{bmatrix} \text{ and } B=\begin{bmatrix}2 & 0 & -1 \\3 & -3 & 4 \end{bmatrix}$$
compute the indicated matrices (if possible). D+BC
Let $$A=\begin{bmatrix}3 & 0 \\ -1 & 5 \end{bmatrix} , B=\begin{bmatrix}4 & -2 & 1 \\ 0 & 2 &3 \end{bmatrix} , C=\begin{bmatrix}1 & 2 \\ 3 & 4 \\ 5 &6 \end{bmatrix} , D=\begin{bmatrix}0 & -3 \\ -2 & 1 \end{bmatrix} , E=\begin{bmatrix}4 & 2 \end{bmatrix} , F=\begin{bmatrix}-1 \\ 2 \end{bmatrix}$$
compute the indicated matrices (if possible). B - C
Let
$$A=\begin{bmatrix}3 & 0 \\-1 & 5 \end{bmatrix} , B=\begin{bmatrix}4 & -2&1 \\0 & 2&3 \end{bmatrix} , C=\begin{bmatrix}1 & 2 \\3 & 4\\5&6 \end{bmatrix}, D=\begin{bmatrix}0 & -3 \\-2 & 1 \end{bmatrix},E=\begin{bmatrix}4 & 2 \end{bmatrix},F=\begin{bmatrix}-1 \\2 \end{bmatrix}$$
compute the indicated matrices . FE
$$A=\begin{bmatrix}3 & 0 \\-1 & 5 \end{bmatrix} , B=\begin{bmatrix}4 & -2&1 \\0 & 2&3 \end{bmatrix} , C=\begin{bmatrix}1& 2 \\3 & 4\\5&6 \end{bmatrix} , D=\begin{bmatrix}0 & -3 \\-2 & 1 \end{bmatrix} , E=\begin{bmatrix}4 & 2 \end{bmatrix} ,F=\begin{bmatrix}-1 \\2 \end{bmatrix}$$
Simplify
1)$$\begin{bmatrix}4 & 5 \end{bmatrix}+\begin{bmatrix}6 & -4 \end{bmatrix}$$
2)$$\begin{bmatrix}4 & -1\\3&3\\-5&-4 \end{bmatrix}-\begin{bmatrix}4 & -2\\3&6\\-5&-6 \end{bmatrix}$$
3)$$\begin{bmatrix}4 & -1\\6&-3 \end{bmatrix}+\begin{bmatrix}5 & -6\\5&-5 \end{bmatrix}-\begin{bmatrix}-2 & 0\\-2&-6 \end{bmatrix}$$
Solve for x and y
$$\begin{bmatrix}-10 & -4\\x&-1 \end{bmatrix}+\begin{bmatrix}-5 & 8\\y&-10 \end{bmatrix}=\begin{bmatrix}-15 & x\\16&-11 \end{bmatrix}$$
Matrices A and B are defined as follows, Find the product: BA $$A=\begin{bmatrix}1 & 2 & 3 \end{bmatrix} B= \begin{bmatrix}4 \\ 5 \\ 6 \end{bmatrix}$$
$$\begin{bmatrix}2 & -1&4 \\g & 0&3\\2&h&0 \end{bmatrix} \times \begin{bmatrix}-1 & 5 \\4&f\\-3&1 \end{bmatrix}=\begin{bmatrix}i & 24 \\-16&-4\\4&e \end{bmatrix}$$