Matrices C and D are shown belowC=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=0.1b)a=0.1 , b=0.5c)a=-0.5 , b=-0.1

Question

Matrices C and D are shown below
$C = [\begin{matrix} 2 & 1 & 0 \\ 0 & 3 & 4 \\ 0 & 2 & 1 \end{matrix}], D = [\begin{matrix} a & b & - 0.4 \\ 0 & - 0.2 & 0.8 \\ 0 & 0.4 & - 0.6 \end{matrix}]$
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

Answer 1

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