Question

Matrices C and D are shown below:

C:
| a b |
| 0 4 |

D:
| 0 3 |
| 2 0 |

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
d) a = -0.1, b = 0.5

Answer 1

The values of a and b that make the equation CD - I true are given by option c (a = 0.5, b = -0.1). Thus the correct option is C.

Step-by-step explanation:

To determine the values of \(a\) and \(b\) that satisfy the equation CD - I, we first need to compute the product CD and then subtract the identity matrix I.

The matrix product CD is obtained by multiplying matrices C and D:

`\[ CD = \begin{bmatrix} a \cdot 0 + b \cdot 2 & a \cdot 3 + b \cdot 0 \\ 0 \cdot 0 + 4 \cdot 2 & 0 \cdot 3 + 4 \cdot 0 \end{bmatrix} = \begin{bmatrix} 2b & 3a \\ 8 & 0 \end{bmatrix} \]`

Now, subtracting the identity matrix I:

`\[ CD - I = \begin{bmatrix} 2b & 3a \\ 8 & 0 \end{bmatrix} - \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 2b - 1 & 3a \\ 8 & -1 \end{bmatrix} \]`

For the equation CD - I to be true, we set each element of the resulting matrix equal to zero. Solving the system of equations, we find that (a = 0.5) and (b = -0.1).

In conclusion, option c (a = 0.5, b = -0.1) provides the values for a and b that make the equation CD - I true. Thus the correct option is C.