Question 1

Provide an appropriate response.
-Let $P$ be a $5 \times 5$ matrix that has a multiplicative inverse. Which of the following statements are false? (Let 0 represent the $5 \times 5$ zero matrix.)
(i) $0 \cdot \mathrm { P } = 0$
(ii) $\mathrm { PP } ^ { - 1 } = 0$
(iii) $\mathrm { P } ^ { - 1 } \mathrm { P } = \mathrm { I } _ { 5 }$
(iv) $\mathrm { PP } ^ { - 1 } = \mathrm { P }$
A) (i), (iii)
B) (ii), (iv)
C) (iv)
D) (ii)

Accepted Answer

Statement (ii) is false because $PP^{-1} = I_5$, not 0. Statement (iv) is false because $PP^{-1} = I_5$, not P.

Question 2

Provide an appropriate response.
-Let M be a $3 \times 3$ matrix. Which of the following statements are true?
(i) $\mathrm { I } _ { 3 } \mathrm { M } = \mathrm { M }$
(ii) $\mathrm { I } _ { 3 } + \mathrm { M } = \mathrm { M }$
(iii) $\mathrm { MI } _ { 3 } = \mathrm { I } _ { 3 }$
(iv) $\quad \mathrm { MI } _ { 3 } = \mathrm { M }$
A) (i), (iii), (iv)
B) (i), (ii), (iii), (iv)
C) (ii), (iii)
D) (i), (iv)

Accepted Answer

(i) is true because multiplying any matrix by the identity matrix of appropriate size on the left yields the original matrix. (iv) is also true because multiplying any matrix by the identity matrix of appropriate size on the right yields the original matrix. (ii) is false because adding the identity matrix to M alters its elements, not keeping it the same. (iii) is incorrectly stated as it implies M equals the identity matrix after multiplication, which is not generally true unless M is specifically the inverse of the identity matrix, which is itself.

Question 3

Provide an appropriate response.
-Let $\mathrm { A } = \left[ \begin{array} { l l } \mathrm { c } & 0 \ 0 & \mathrm {~b} \end{array} \right]$, where $\mathrm { c }$ and $\mathrm { b }$ are nonzero constants. Find $\mathrm { A } ^ { - 1 }$.
A)
$$\left[ \begin{array} { l l } 
\frac { 1 } { c } & 0 \
0 & \frac { 1 } { b }
\end{array} \right]$$

B)
$\left[\begin{array}{ll}
0 & \mathrm{c} \
\mathrm{b} & 0
\end{array}\right]$

C)
$\left[\begin{array}{l}
\frac{1}{c} 1 \
1 \frac{1}{b}
\end{array}\right]$

D)

$\left[\begin{array}{ll}
\mathrm{c} & 0 \
0 & \mathrm{~b}
\end{array}\right]$

Accepted Answer

The inverse of a diagonal matrix $\left[ \begin{array} { l l } c & 0 \ 0 & b \end{array} \right]$ is obtained by taking the reciprocal of each nonzero diagonal element, resulting in $\left[ \begin{array} { l l } \frac { 1 } { c } & 0 \ 0 & \frac { 1 } { b } \end{array} \right]$.

Question 4

Provide an appropriate response.
-Let $A = \left[ \begin{array} { r r } - 4 & - 4 \ - 6 & 3 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 1 & 4 \ 5 & - 3 \end{array} \right]$
Does the matrix $A + B$ have an inverse?
A) Yes
B) No

Accepted Answer

The answer of Provide an appropriate response.
-Let \(A = \left[...

Question 5

Provide an appropriate response.
-Let $\mathrm { A } = \left[ \begin{array} { r r } - 3 & - 6 \ 3 & 5 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } - 3 & - 6 \ 4 & 5 \end{array} \right]$.
Does the matrix A - B have an inverse?
A) $\mathrm { Yes }$
B) $\mathrm { No }$

Accepted Answer

The matrix $A - B = \left[ \begin{array} { r r } 0 & 0 \ -1 & 0 \end{array} \right]$. This matrix does not have an inverse because its determinant is 0, indicating it is singular.

Question 6

Provide an appropriate response.
-Does a square matrix with a row of all zeros have an inverse?
A) Yes
B) $\mathrm { No }$

Accepted Answer

A square matrix with a row of all zeros cannot have an inverse because its determinant is zero, indicating it is singular and not invertible.

Question 7

Provide an appropriate response.
-Consider the matrix
$$A = \left[ \begin{array} { r r r } 
2 & 9 & 3 \
4 & 10 & 0 \
- 2 & - 7 & 0
\end{array} \right]$$
Without using a calculator, determine whether $\mathrm { A } ^ { - 1 }$ exists. Explain your reasoning.

Accepted Answer

The answer of Provide an appropriate response.
-Consider the matrix
\[A =...

Quiz 10: Systems and Matrices

Provide an appropriate response. -Let $\mathrm { A } = \left[ \begin{array} { l l } \mathrm { c } & 0 \\ 0 & \mathrm {~b} \end{array} \right]$ , where $\mathrm { c }$ and $\mathrm { b }$ are nonzero constants. Find $\mathrm { A } ^ { - 1 }$ .

Provide an appropriate response. -Let $A = \left[ \begin{array} { r r } - 4 & - 4 \\ - 6 & 3 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 1 & 4 \\ 5 & - 3 \end{array} \right]$ Does the matrix $A + B$ have an inverse?

Provide an appropriate response. -Let $\mathrm { A } = \left[ \begin{array} { r r } - 3 & - 6 \\ 3 & 5 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } - 3 & - 6 \\ 4 & 5 \end{array} \right]$ . Does the matrix A - B have an inverse?

Provide an appropriate response. -Does a square matrix with a row of all zeros have an inverse?

Provide an appropriate response. -Consider the matrix $A = \left[ \begin{array} { r r r } 2 & 9 & 3 \\4 & 10 & 0 \\- 2 & - 7 & 0\end{array} \right]$ Without using a calculator, determine whether $\mathrm { A } ^ { - 1 }$ exists. Explain your reasoning.

Quiz 10: Systems and Matrices

Provide an appropriate response. -Let A=[c00 b]\mathrm { A } = \left[ \begin{array} { l l } \mathrm { c } & 0 \\ 0 & \mathrm {~b} \end{array} \right]A=[c0​0 b​] , where c\mathrm { c }c and b\mathrm { b }b are nonzero constants. Find A−1\mathrm { A } ^ { - 1 }A−1 .

Provide an appropriate response. -Let A=[−4−4−63]A = \left[ \begin{array} { r r } - 4 & - 4 \\ - 6 & 3 \end{array} \right]A=[−4−6​−43​] and B=[145−3]B = \left[ \begin{array} { r r } 1 & 4 \\ 5 & - 3 \end{array} \right]B=[15​4−3​] Does the matrix A+BA + BA+B have an inverse?

Provide an appropriate response. -Does a square matrix with a row of all zeros have an inverse?

Provide an appropriate response. -Let $\mathrm { A } = \left[ \begin{array} { l l } \mathrm { c } & 0 \\ 0 & \mathrm {~b} \end{array} \right]$ , where $\mathrm { c }$ and $\mathrm { b }$ are nonzero constants. Find $\mathrm { A } ^ { - 1 }$ .

Provide an appropriate response. -Let $A = \left[ \begin{array} { r r } - 4 & - 4 \\ - 6 & 3 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 1 & 4 \\ 5 & - 3 \end{array} \right]$ Does the matrix $A + B$ have an inverse?