$\C=\left[\begin{array}{lll}2 & 1 & 0 \\ 0 & 3 & 4 \\ 0 & 2 & 1\end{array}\right] \quad D=\left[\begin{array}{ccc}a & b & -0.4 \\ 0 & -0.2 & 0.8 \\ 0 & 0.4 & -0.6\end{array}\right]$
What values of $a$ and $b$ will make the equation $C D=I$ true?
A. $a=0.5$, $b=0.1$
B. $a=0.1$, $b=0.5$
C. $a=-0.5$, $b=-0.1$
D. $a=-0.1$, $b=-0.5$
Answer (1)
Calculate the matrix product CD.
Set the resulting matrix equal to the identity matrix I.
Solve the resulting system of equations for a and b.
The values of a and b are a = 0.5 , b = 0.1 .
Explanation
Problem Setup We are given two matrices, C and D , and we want to find the values of a and b such that C D = I , where I is the identity matrix.
Matrices Definition The given matrices are:
C = [ 2 1 0 0 3 4 0 2 1 ] D = [ a b − 0.4 0 − 0.2 0.8 0 0.4 − 0.6 ]
We need to find a and b such that their product equals the identity matrix:
I = [ 1 0 0 0 1 0 0 0 1 ]
Calculating CD Let's compute the matrix product C D :
C D = [ 2 1 0 0 3 4 0 2 1 ] [ a b − 0.4 0 − 0.2 0.8 0 0.4 − 0.6 ] = [ 2 a 2 b − 0.2 0 0 − 0.6 + 1.6 2.4 − 2.4 0 − 0.4 + 0.4 1.6 − 0.6 ] = [ 2 a 2 b − 0.2 0 0 1 0 0 0 1 ]
Setting CD = I Now, we set C D equal to the identity matrix I :
[ 2 a 2 b − 0.2 0 0 1 0 0 0 1 ] = [ 1 0 0 0 1 0 0 0 1 ]
From this, we get the following equations:
2 a = 1
2 b − 0.2 = 0
Solving for a and b Solving for a and b :
From equation 1, we have:
2 a = 1 ⟹ a = 2 1 = 0.5
From equation 2, we have:
2 b − 0.2 = 0 ⟹ 2 b = 0.2 ⟹ b = 2 0.2 = 0.1
Final Answer Therefore, the values of a and b that make C D = I true are a = 0.5 and b = 0.1 .
Examples
Matrix multiplication is used in computer graphics to perform transformations such as scaling, rotation, and translation of objects. The matrices C and D could represent transformations, and the equation CD = I means that D is the inverse transformation of C, which undoes the transformation performed by C. This is useful in applications where you need to revert an object back to its original state after applying a series of transformations.
