Question

# Matrices C and D are shown below C=begin{bmatrix}2&1&0 0&3&40&2&1 end{bmatrix},D=begin{bmatrix}a & b&-0.4 0&-0.2&0.80&0.4&-0.6 end{bmatrix} What values of a and b will make the equation CD=I true? a)a=0.5 , b=0.1 b)a=0.1 , b=0.5 c)a=-0.5 , b=-0.1

Matrices
Matrices C and D are shown below
C=\begin{bmatrix}2&1&0 \\0&3&4\\0&2&1 \end{bmatrix},D=\begin{bmatrix}a & b&-0.4 \\0&-0.2&0.8\\0&0.4&-0.6 \end{bmatrix}
What values of a and b will make the equation CD=I true?
a)a=0.5 , b=0.1
b)a=0.1 , b=0.5
c)a=-0.5 , b=-0.1

2021-02-14

Step 1
The given matrices are,
$$C=\begin{bmatrix}2&1&0 \\0&3&4\\0&2&1 \end{bmatrix}\text{ and }D=\begin{bmatrix}a & b&-0.4 \\0&-0.2&0.8\\0&0.4&-0.6 \end{bmatrix}$$
Step 2
Now multiply the matrices C and D as shown below.
$$CD=\begin{bmatrix}2&1&0 \\0&3&4\\0&2&1 \end{bmatrix}\begin{bmatrix}a & b&-0.4 \\0&-0.2&0.8\\0&0.4&-0.6 \end{bmatrix}$$
$$=\begin{bmatrix}2a+0+0 & 2b-0.2+0&-0.8+0.8+0 \\0+0+0 & 0-0.6+1.6&0+2.4-2.4\\0+0+0&0-0.4+0.4&0+1.6-0.6 \end{bmatrix}$$
$$=\begin{bmatrix}2a & 2b-0.2&0 \\0 & 1&0\\0&0&1 \end{bmatrix}$$
Step 3
Now equate the matrix CD to the identity matrix I and obtain the values of a and b as follows.
CD=I
$$\begin{bmatrix}2a & 2b-0.2&0 \\0 & 1&0\\0&0&1 \end{bmatrix}=\begin{bmatrix}1 & 0&0 \\0 & 1&0\\0&0&1 \end{bmatrix}$$
Two matrices are equal , only if the corresponding elements are equal.
$$2a=1$$
$$\Rightarrow a=0.5$$
and
$$2b-0.2=0$$
$$\Rightarrow b=0.1$$
Step 4
Therefore, the CD = I is true for a = 0.5 and b =0.1.