# Consider the following two matrices. Why can't the product of the following two matrices be found? A=begin{bmatrix}-1 & 2&3 4 & 0&5 end{bmatrix} text{ and } B=begin{bmatrix}5 & 2 7 & -8 end{bmatrix}

Question
Matrices
Consider the following two matrices. Why can't the product of the following two matrices be found? $$A=\begin{bmatrix}-1 & 2&3 \\4 & 0&5 \end{bmatrix} \text{ and } B=\begin{bmatrix}5 & 2 \\7 & -8 \end{bmatrix}$$

2021-02-26
Step 1
A=\begin{bmatrix}-1 & 2&3 \\4 & 0&5 \end{bmatrix} \text{ and } B=\begin{bmatrix}5 & 2 \\7 & -8 \end{bmatrix}
We explain why matrix product is not possible.
Step 2
If A is a matrix of order $$m \times n$$ and B is a matrix of order $$p \times q$$
The matrix product AB is possible if
n=p
Here A is a matrix of order $$2 \times 3 (m \times n)$$
B is a matrix of order $$2 \times 2 (p \times q)$$
Here m=2,n=3,p=2,q=2
Here $$n=3 \neq p=2$$
Since $$n \neq p$$ so matrix product not possible

### Relevant Questions

Given the two matrices,
$$A=\begin{bmatrix}1 & 2&3 \\1 & 1&2\\0&1&2 \end{bmatrix} \text{ and } B=\begin{bmatrix}1 & 1&1 \\2 & 1&2\\3&1&2 \end{bmatrix}$$
(a) Find det A, det B , det(AB) , det(BA) , det(5A) , $$det A^T$$ and $$det(B^6)$$
(c) Find $$A^{-1} and B^{-1} using the adjoint matrices you found in part (b) asked 2021-03-09 Use the graphing calculator to solve if possible A=\begin{bmatrix}1 & 0&5 \\1 & -5&7\\0&3&-4 \end{bmatrix}\\ B=\begin{bmatrix}3 & -5&3 \\2&3&1\\4&1&-3\end{bmatrix}\\ C=\begin{bmatrix}5 & 2&3 \\2& -1&0 \end{bmatrix}\\ D=\begin{bmatrix}5 \\-3\\4 \end{bmatrix} Find the value in row 2 column 3 of AB-3B asked 2020-12-16 Consider the matrices \(A=\begin{bmatrix}1 & -1 \\0 & 1 \end{bmatrix},B=\begin{bmatrix}2 & 3 \\1 & 5 \end{bmatrix},C=\begin{bmatrix}1 & 0 \\0 & 8 \end{bmatrix},D=\begin{bmatrix}2 & 0 &-1\\1 & 4&3\\5&4&2 \end{bmatrix} \text{ and } F=\begin{bmatrix}2 & -1 &0\\0 & 1&1\\2&0&3 \end{bmatrix}$$
a) Show that A,B,C,D and F are invertible matrices.
b) Solve the following equations for the unknown matrix X.
(i) $$AX^T=BC^3$$
(ii) $$A^{-1}(X-T)^T=(B^{-1})^T$$
(iii) $$XF=F^{-1}-D^T$$
Use the matrices AA and BB below instead of those in your text.
$$A=\begin{bmatrix}-6 & -1 \\ -3 & -4 \end{bmatrix} B=\begin{bmatrix} -1 & 3 \\ -5 & -8 \end{bmatrix}$$ 1) 2A+B=? 2)A-4B=?
Compute the product AB by the definition of the product of​ matrices, where $$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}$$
Determine the product AB
AB=?
find which of the given matrices are nonsingular.
a) $$\begin{bmatrix}1 & 2 &-3 \\-1 & 2&3 \\ 0 &8&0 \end{bmatrix}$$
b)$$\begin{bmatrix}1 & 2 &-3 \\-1 & 2&3 \\ 0 &1&1 \end{bmatrix}$$
c) $$\begin{bmatrix}1 & 1 &2 \\-1 & 3&4 \\ -5 &7&8 \end{bmatrix}$$
d) $$\begin{bmatrix}1 & 1 &4&-1 \\1 & 2&3&2 \\ -1 &3&2&1\\-2&6&12&-4 \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}$$
Consider the three following matrices:
$$A=\begin{bmatrix}1 & 0&0 \\0 & -1&0\\0&0&1 \end{bmatrix} , B=\begin{bmatrix}1 & 0&0 \\0 & 0&0\\0&0&-1 \end{bmatrix} \text{ and } C=\begin{bmatrix}0 & -i&0 \\i & 0&-i\\0&i&0 \end{bmatrix}$$
Calculate the Tr(ABC)
(a)1
(b)2
(c)2i
(d)0
$$A=\begin{bmatrix}0 & 0&1 \\ 0 & 3&0 \\ 1&0 & -10 \end{bmatrix}$$
$$A=\begin{bmatrix}0 & 0&1 \\ 0 & 3&0 \\ 1&0 & -10 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&-10 \\ 0 & 3&0 \\ 0&0 & 1 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&-10 \\ 0 & 1&0 \\ 0&0 & 1 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&0 \\ 0 & 1&0 \\ 0&0 & 1 \end{bmatrix}$$
Write $$A \text{ and } A^{-1}$$ as product of elementary matrices.