[2a+b=4]
[a-2b=-3]
Chapter matrix
Find the values of a,b
Answers
Multiplying a 2 × 3 matrix by a 3 × 4 matrix is possible and it gives a 2 × 4 matrix as the answer.
b) Multiplying a 7 × 1 matrix by a 1 × 2 matrix is okay; it gives a 7 × 2 matrix
c) A 4 × 3 matrix times a 2 × 3 matrix is NOT possible.
How to Multiply 2 Matrices
We use letters first to see what is going on. We'll see a numbers example after.
As an example, let's take a general 2 × 3 matrix multiplied by a 3 × 2 matrix.
\displaystyle{\left[\begin{matrix}{a}&{b}&{c}\\{d}&{e}&{f}\end{matrix}\right]}{\left[\begin{matrix}{u}&{v}\\{w}&{x}\\{y}&{z}\end{matrix}\right]}[
a
d
b
e
c
f
]
⎣
⎡
u
w
y
v
x
z
⎦
⎤
The answer will be a 2 × 2 matrix.
We multiply and add the elements as follows. We work across the 1st row of the first matrix, multiplying down the 1st column of the second matrix, element by element. We add the resulting products. Our answer goes in position a11 (top left) of the answer matrix.
Multiplying matrices - first step
We do a similar process for the 1st row of the first matrix and the 2nd column of the second matrix. The result is placed in position a12.
Multiplying matrices - second step
Now for the 2nd row of the first matrix and the 1st column of the second matrix. The result is placed in position a21.
Multiplying matrices - third step
Finally, we do the 2nd row of the first matrix and the 2nd column of the second matrix. The result is placed in position a22.
Multiplying matrices - fourth step
So the result of multiplying our 2 matrices is as follows:
\displaystyle{\left[\begin{matrix}{a}&{b}&{c}\\{d}&{e}&{f}\end{matrix}\right]}{\left[\begin{matrix}{u}&{v}\\{w}&{x}\\{y}&{z}\end{matrix}\right]}[
a
d
b
e
c
f
]
⎣
⎡
u
w
y
v
x
z
⎦
⎤
\displaystyle={\left[\begin{matrix}{a}{u}+{b}{w}+{c}{y}&{a}{v}+{b}{x}+{c}{z}\\{d}{u}+{e}{w}+{f}{y}&{d}{v}+{e}{x}+{f}{z}\end{matrix}\right]}=[
au+bw+cy
du+ew+fy
av+bx+cz
dv+ex+fz
]