Question 1

In questions determine whether the statement is true or false.
-$$\text { If } \mathbf { A B } = \mathbf { A C } \text {, then } \mathbf { B } = \mathbf { C } \text {. }$$

Accepted Answer

If matrix A is not invertible (i.e., it does not have full rank), then AB = AC does not necessarily imply B = C.

Question 2

Suppose  A  is a  2 X 2  matrix with real number entries such that  AB=BA  for all  2 X 2  matrices. What relationships must exist among the entries of  A  ?

Accepted Answer

The answer of Suppose  A  is a ...

Question 3

Let $\mathbf { A } = \left( \begin{array} { c c } 1 & m \ 0 & 1 \end{array} \right)$Find Aⁿ where n is a positive integer.

Accepted Answer

The answer of Let \(\mathbf { A } = \left(...

Question 4

In questions determine whether the statement is true or false.
-$$\text { If } \mathbf { A } \text { and } \mathbf { B } \text { are } 2 \times 2 \text { matrices, then } \mathbf { A } + \mathbf { B } = \mathbf { B } + \mathbf { A } \text {. }$$

Accepted Answer

Matrix addition is commutative, meaning the order of addition does not affect the result.

Question 5

Suppose  A =$\left( \begin{array} { l l } 
3 & 5 \
2 & 4
\end{array} \right)$ and C= $\left( \begin{array} { l l } 
2 & 1 \
0 & 6
\end{array} \right)$ . Find a matrix B  such that AB=C  or prove that no such matrix exists.

Accepted Answer

The answer of Suppose  A =\(\left( \begin{array} { l...

Question 6

Suppose A=$$\left( \begin{array} { l l l } 
1 & 0 & 1 \
0 & 1 & 1 \
1 & 1 & 0
\end{array} \right)$$ and  B=$$\left( \begin{array} { l l l } 
0 & 1 & 0 \
0 & 1 & 1 \
1 & 0 & 0
\end{array} \right)$$ . Find
(a) the join of  A and  B .
(b) the meet of  A  and B .
(c) the Boolean product of  A  and  B .

Accepted Answer

The answer of Suppose A=\[\left( \begin{array} { l l l...

Question 7

Suppose B=$\left( \begin{array} { l l } 
6 & 2 \
3 & 1
\end{array} \right)$  and C=$\left( \begin{array} { l l } 
2 & 1 \
0 & 6
\end{array} \right)$ .  Find a matrix  A  such that  AB=C  or prove that no such matrix exists.

Accepted Answer

The answer of Suppose B=\(\left( \begin{array} { l l }...

Question 8

In questions determine whether the statement is true or false.
-$$\text { If } \mathbf { A } \text { and } \mathbf { B } \text { are } 2 \times 2 \text { matrices such that } \mathbf { A B } = \left( \begin{array} { l l } 
0 & 0 \
0 & 0
\end{array} \right) \text {, then } \mathbf { A } = \left( \begin{array} { l l } 
0 & 0 \
0 & 0
\end{array} \right) \text { or } \mathbf { B } = \left( \begin{array} { l l } 
0 & 0 \
0 & 0
\end{array} \right) \text {. }$$

Accepted Answer

The product of two non-zero matrices can be the zero matrix if they are singular and their null spaces align appropriately.

Question 9

In questions determine whether the statement is true or false.
-$$\text { If } \mathbf { A } = \left( \begin{array} { l l } 
3 & 5 \
1 & 2
\end{array} \right) , \text { then } \mathbf { A } ^ { - 1 } = \left( \begin{array} { c c } 
2 & 5 \
1 & - 3
\end{array} \right)$$

Accepted Answer

The inverse of a 2x2 matrix $\mathbf{A} = \begin{pmatrix} a & b \ c & d \end{pmatrix}$ is $\mathbf{A}^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}$. For $\mathbf{A} = \begin{pmatrix} 3 & 5 \ 1 & 2 \end{pmatrix}$, the determinant is $3(2) - 5(1) = 1$, so $\mathbf{A}^{-1} = \begin{pmatrix} 2 & -5 \ -1 & 3 \end{pmatrix}$.

Question 10

In questions determine whether the statement is true or false.
-$$\text { If } \mathbf { A } = \left( \begin{array} { c c } 
1 & 3 \
- 5 & 2
\end{array} \right) , \text { then } \mathbf { A } ^ { 2 } = \left( \begin{array} { c c } 
1 & 9 \
25 & 4
\end{array} \right)$$

Accepted Answer

Matrix multiplication is not element-wise; $\mathbf{A}^2$ is calculated by multiplying $\mathbf{A}$ by itself, not squaring each element.

Question 11

In questions determine whether the statement is true or false.
-$$\text { If } \mathbf { A } \text { is a } 6 \times 4 \text { matrix and } \mathbf { B } \text { is a } 4 \times 5 \text { matrix, then } \mathbf { A B } \text { has } 16 \text { entries. }$$

Accepted Answer

The resulting matrix AB will be a 6x5 matrix, which has 30 entries, not 16.

Question 12

Suppose B = $\left( \begin{array} { l l } 
3 & 5 \
2 & 4
\end{array} \right)$ and C = $\left( \begin{array} { l l } 
2 & 1 \
0 & 6
\end{array} \right)$ .  Find a matrix A such that AB=C  or prove that no such matrix exists.

Accepted Answer

The answer of Suppose B = \(\left( \begin{array} { l...

Question 13

In questions determine whether the statement is true or false.
-$$\mathbf { A } \mathbf { B } = \mathbf { B } \mathbf { A } \text { for all } 2 \times 2 \text { matrices } \mathbf { A } \text { and } \mathbf { B } \text {. }$$

Accepted Answer

Matrix multiplication is not commutative, so $\mathbf{AB}$ does not necessarily equal $\mathbf{BA}$ for all matrices $\mathbf{A}$ and $\mathbf{B}$.

Question 14

Suppose A  is a  6 X 8  matrix, B  is an  8 X 5  matrix, and  C  is a  5 X 9  matrix. Find the number of rows, the number of columns, and the number of entries in A(BC).

Accepted Answer

The answer of Suppose A  is a  6...

Quiz 2: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

In questions determine whether the statement is true or false. - $\text { If } \mathbf { A B } = \mathbf { A C } \text {, then } \mathbf { B } = \mathbf { C } \text {. }$

Suppose A is a 2 X 2 matrix with real number entries such that AB=BA for all 2 X 2 matrices. What relationships must exist among the entries of A ?

Let $\mathbf { A } = \left( \begin{array} { c c } 1 & m \\0 & 1\end{array} \right)$ Find Aⁿ where n is a positive integer.

In questions determine whether the statement is true or false. - $\text { If } \mathbf { A } \text { and } \mathbf { B } \text { are } 2 \times 2 \text { matrices, then } \mathbf { A } + \mathbf { B } = \mathbf { B } + \mathbf { A } \text {. }$

Suppose A = $\left( \begin{array} { l l } 3 & 5 \\2 & 4\end{array} \right)$ and C= $\left( \begin{array} { l l } 2 & 1 \\0 & 6\end{array} \right)$ . Find a matrix B such that AB=C or prove that no such matrix exists.

Suppose A= $\left( \begin{array} { l l l } 1 & 0 & 1 \\0 & 1 & 1 \\1 & 1 & 0\end{array} \right)$ and B= $\left( \begin{array} { l l l } 0 & 1 & 0 \\0 & 1 & 1 \\1 & 0 & 0\end{array} \right)$ . Find (a) the join of A and B . (b) the meet of A and B . (c) the Boolean product of A and B .

Suppose B= $\left( \begin{array} { l l } 6 & 2 \\3 & 1\end{array} \right)$ and C= $\left( \begin{array} { l l } 2 & 1 \\0 & 6\end{array} \right)$ . Find a matrix A such that AB=C or prove that no such matrix exists.

In questions determine whether the statement is true or false. - $\text { If } \mathbf { A } = \left( \begin{array} { l l } 3 & 5 \\1 & 2\end{array} \right) , \text { then } \mathbf { A } ^ { - 1 } = \left( \begin{array} { c c } 2 & 5 \\1 & - 3\end{array} \right)$

In questions determine whether the statement is true or false. - $\text { If } \mathbf { A } = \left( \begin{array} { c c } 1 & 3 \\- 5 & 2\end{array} \right) , \text { then } \mathbf { A } ^ { 2 } = \left( \begin{array} { c c } 1 & 9 \\25 & 4\end{array} \right)$

In questions determine whether the statement is true or false. - $\text { If } \mathbf { A } \text { is a } 6 \times 4 \text { matrix and } \mathbf { B } \text { is a } 4 \times 5 \text { matrix, then } \mathbf { A B } \text { has } 16 \text { entries. }$

Suppose B = $\left( \begin{array} { l l } 3 & 5 \\2 & 4\end{array} \right)$ and C = $\left( \begin{array} { l l } 2 & 1 \\0 & 6\end{array} \right)$ . Find a matrix A such that AB=C or prove that no such matrix exists.

In questions determine whether the statement is true or false. - $\mathbf { A } \mathbf { B } = \mathbf { B } \mathbf { A } \text { for all } 2 \times 2 \text { matrices } \mathbf { A } \text { and } \mathbf { B } \text {. }$

Suppose A is a 6 X 8 matrix, B is an 8 X 5 matrix, and C is a 5 X 9 matrix. Find the number of rows, the number of columns, and the number of entries in A(BC).

Quiz 2: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

In questions determine whether the statement is true or false. - If AB=AC, then B=C. \text { If } \mathbf { A B } = \mathbf { A C } \text {, then } \mathbf { B } = \mathbf { C } \text {. } If AB=AC, then B=C.

Suppose A is a 2 X 2 matrix with real number entries such that AB=BA for all 2 X 2 matrices. What relationships must exist among the entries of A ?

Let A=(1m01)\mathbf { A } = \left( \begin{array} { c c } 1 & m \\0 & 1\end{array} \right)A=(10​m1​) Find An where n is a positive integer.

Suppose A = (3524)\left( \begin{array} { l l } 3 & 5 \\2 & 4\end{array} \right)(32​54​) and C= (2106)\left( \begin{array} { l l } 2 & 1 \\0 & 6\end{array} \right)(20​16​) . Find a matrix B such that AB=C or prove that no such matrix exists.

Suppose B= (6231)\left( \begin{array} { l l } 6 & 2 \\3 & 1\end{array} \right)(63​21​) and C= (2106)\left( \begin{array} { l l } 2 & 1 \\0 & 6\end{array} \right)(20​16​) . Find a matrix A such that AB=C or prove that no such matrix exists.

Suppose B = (3524)\left( \begin{array} { l l } 3 & 5 \\2 & 4\end{array} \right)(32​54​) and C = (2106)\left( \begin{array} { l l } 2 & 1 \\0 & 6\end{array} \right)(20​16​) . Find a matrix A such that AB=C or prove that no such matrix exists.

In questions determine whether the statement is true or false. - AB=BA for all 2×2 matrices A and B. \mathbf { A } \mathbf { B } = \mathbf { B } \mathbf { A } \text { for all } 2 \times 2 \text { matrices } \mathbf { A } \text { and } \mathbf { B } \text {. }AB=BA for all 2×2 matrices A and B.