Efan Halliday

2021-02-13

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

Aniqa O'Neill

Step 1
The given matrices are,

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

Jeffrey Jordon

Answer is given below (on video)

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