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Linear Algebra : Linear Algebra

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4 Diagnostic Tests 108 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #3 : Matrices

Compute $A\cdot B$ where,

$A=\begin{bmatrix} 4& 5 \end{bmatrix}$

$B=\begin{bmatrix} 0& 4&2 \\ 7&0 &1 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 0&35 \\ 16& 13 \end{bmatrix}$

$\begin{bmatrix} 35\\ 16 \\ 13 \end{bmatrix}$

$\begin{bmatrix} 35&16 & 13 \end{bmatrix}$

$\begin{bmatrix} 13&16 & 35 \end{bmatrix}$

Not Possible

Correct answer:

$\begin{bmatrix} 35&16 & 13 \end{bmatrix}$

Explanation:

Since the number of columns in the first matrix equals the number of rows in the second matrix, we know that these matrices can be multiplied together. To determine the dimensions of the product matrix, we take the number of rows in the first matrix and the number of columns in the second matrix. For this example, our product matrix will have dimensions of (1x3). The product matrix equals, $\begin{bmatrix} (4\cdot 0+5\cdot 7)&(4\cdot 4 +5\cdot 0)&(4\cdot 2 +5\cdot 1) \end{bmatrix}$

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Example Question #4 : Matrices

Compute $A\cdot B$ where,

$A=\begin{bmatrix} 0&1 \\ 1&0 \end{bmatrix}$

$B=\begin{bmatrix} 7\\ 5 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 5\\ 7 \end{bmatrix}$

$\begin{bmatrix} 12 \end{bmatrix}$

$\begin{bmatrix} 7\\ 5 \end{bmatrix}$

$\begin{bmatrix} 7&5 \\ 5& 7 \end{bmatrix}$

Not Possible

Correct answer:

$\begin{bmatrix} 5\\ 7 \end{bmatrix}$

Explanation:

Since the number of columns in the first matrix equals the number of rows in the second matrix, we know that these matrices can be multiplied together. To determine the dimensions of the product matrix, we take the number of rows in the first matrix and the number of columns in the second matrix. For this example, our product matrix will have dimensions of (2x1). The product matrix equals, $\begin{bmatrix} (0\cdot 7+1\cdot 5)\\(1\cdot 7+0\cdot 5)\end{bmatrix}$

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Example Question #691 : Linear Algebra

Compute $A\cdot B$ where,

$A=\begin{bmatrix} 1\\ 2\\ 5 \end{bmatrix}$

$B=\begin{bmatrix} 1\\ 5 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 1\\ 25 \end{bmatrix}$

$\begin{bmatrix} 2&25 \\ 7& 3 \end{bmatrix}$

Not Possible

$\begin{bmatrix} 2& 25 \end{bmatrix}$

$\begin{bmatrix} 2\\ 7 \\25 \end{bmatrix}$

Correct answer:

Not Possible

Explanation:

Since the number of columns in the first matrix does not equal the number of rows in the second matrix, you cannot multiply these two matrices.

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Example Question #8 : Matrix Matrix Product

Compute $A\cdot B$ where,

$A=\begin{bmatrix} 0&0 \\ 1&1 \end{bmatrix}$

$B=\begin{bmatrix} 1&1 \\ 0& 0 \end{bmatrix}$

Possible Answers:

Not Possible

$\begin{bmatrix} 0&0 \\ 1&1 \end{bmatrix}$

$\begin{bmatrix} 0&1 \\ 1& 0 \end{bmatrix}$

$\begin{bmatrix} 1&1 \\ 0&0 \end{bmatrix}$

$\begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 0&0 \\ 1&1 \end{bmatrix}$

Explanation:

Since the number of columns in the first matrix equals the number of rows in the second matrix, we know that these matrices can be multiplied together. To determine the dimensions of the product matrix, we take the number of rows in the first matrix and the number of columns in the second matrix. For this example, our product matrix will have dimensions of (2x2). The product matrix equals, $\begin{bmatrix} (0\cdot 1+0\cdot 0)&(0\cdot 1 +0\cdot 0)\\(1\cdot 1 +1\cdot 0)&(1\cdot 1 +1\cdot 0) \end{bmatrix}$

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Example Question #2 : Matrix Matrix Product

Compute $A\cdot B$ where,

$A=\begin{bmatrix} 12 \end{bmatrix}$

$B=\begin{bmatrix} 4&5 &1 &0 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 48\\ 60\\ 12\\ 0 \end{bmatrix}$

$\begin{bmatrix} 0\\ 12\\ 60\\ 48 \end{bmatrix}$

$\begin{bmatrix} 48& 60&12 & 0 \end{bmatrix}$

Not Possible

$\begin{bmatrix} 16&17 & 13& 12 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 48& 60&12 & 0 \end{bmatrix}$

Explanation:

Since the number of columns in the first matrix equals the number of rows in the second matrix, we know that these matrices can be multiplied together. To determine the dimensions of the product matrix, we take the number of rows in the first matrix and the number of columns in the second matrix. For this example, our product matrix will have dimensions of (1x4). The product matrix equals, $\begin{bmatrix} 12\cdot 4&12\cdot 5&12\cdot 1 &12\cdot 0 \end{bmatrix}$

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Example Question #6 : Matrices

Compute $A\cdot B$ where,

$A=\begin{bmatrix} 0&1 &4 \\ 4&0 &7 \\ 2& 5& 0 \end{bmatrix}$

$B=\begin{bmatrix} 1\\ 2\\ 0 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 0&1 &4 \\ 8&0 &14 \\ 0&0 &0 \end{bmatrix}$

$\begin{bmatrix} 2& 4&12 \end{bmatrix}$

Not Possible

$\begin{bmatrix} 25 \end{bmatrix}$

$\begin{bmatrix} 2\\ 4\\ 12 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 2\\ 4\\ 12 \end{bmatrix}$

Explanation:

Since the number of columns in the first matrix equals the number of rows in the second matrix, we know that these matrices can be multiplied together. To determine the dimensions of the product matrix, we take the number of rows in the first matrix and the number of columns in the second matrix. For this example, our product matrix will have dimensions of (3x1). The product matrix equals, $\begin{bmatrix} (0\cdot 1+1\cdot 2+4\cdot 0)\\(4\cdot 1+0\cdot 2+7\cdot 0)\\(2\cdot 1+5\cdot 2+0\cdot 0)\end{bmatrix}$

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Example Question #11 : Matrices

Compute $A\cdot B$ , where

$A=\begin{bmatrix}8&4 \\ 2&4 \\ 6&1 \end{bmatrix}$ $b=\begin{bmatrix}4&2 \\ 4&9 \\ 2&1 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 16\\-28 \\ 11 \end{bmatrix}$

$\begin{bmatrix} 32&16 &8 \\ 8&36 &24 \end{bmatrix}$

Not Possible

$\begin{bmatrix} 32&8 \\ 8&36 \\ 12&1 \end{bmatrix}$

$\begin{bmatrix}8&32 \\ 36&8 \\ 1&12 \end{bmatrix}$

Correct answer:

Not Possible

Explanation:

In order to be able to multiply matrices, the number of columns of the 1st matrix must equal the number of rows in the second matrix. Here, the first matrix has dimensions of (3x2). This means it has three rows and two columns. The second matrix has dimensions of (3x2), also three rows and two columns. Since $\tiny \tiny \small 2\neq3$ , we cannot multiply these two matrices together

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Example Question #11 : Matrices

$C = \begin{bmatrix} 4& 0.7\\ 7& -0.4 \end{bmatrix}$ .

Evaluate $C^{T}C$ .

Possible Answers:

$\begin{bmatrix} 0.49 & 0 \\0 & 16\end{bmatrix}$

$\begin{bmatrix}0.65 & 0 \\ 0 & 65 \end{bmatrix}$

$\begin{bmatrix} 16 & 0 \\0 & 0.49\end{bmatrix}$

$\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$

$\begin{bmatrix}65 & 0 \\ 0 & 0.65 \end{bmatrix}$

Correct answer:

$\begin{bmatrix}65 & 0 \\ 0 & 0.65 \end{bmatrix}$

Explanation:

The transpose of a matrix switches the rows and the columns. Therefore, the first column of $C^{T}$ has the same entries, in order, as the first row of , and so forth. Since

$C = \begin{bmatrix} 4& 0.7\\ 7& -0.4 \end{bmatrix}$ ,

$C^{T} = \begin{bmatrix} 4& 7\\ 0.7& -0.4 \end{bmatrix}$ .

The entry in column , row of the product $C^{T}C$ is the product of row of $C^{T}$ and column of - the sum of the products of the numbers that appear in the corresponding positions of the row and the column. $C^{T}C$ can consequently be calculated as follows:

$C^{T}C$

$= \begin{bmatrix} 4& 7\\ 0.7& -0.4 \end{bmatrix} \begin{bmatrix} 4& 0.7\\ 7& -0.4 \end{bmatrix}$

$= \begin{bmatrix} 4 (4) + 7 (7) & 4 (0.7) + 7 (-0.4)\\ 0.7 (4) + (-0.4) (7) & 0.7 (0.7) + (-0.4) (-0.4) \end{bmatrix}$

$= \begin{bmatrix}16+49 & 2.8 + (-2.8) \\ 2.8 + (-2.8) & 0.49 + 0.16 \end{bmatrix}$

$= \begin{bmatrix}65 & 0 \\ 0 & 0.65 \end{bmatrix}$ ,

the correct choice.

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Example Question #11 : Matrix Matrix Product

$C = \begin{bmatrix} 3&0 & 6\\ -7&1 &2 \end{bmatrix}$

Evaluate $C^{T}C$ .

Possible Answers:

$\begin{bmatrix} 9 & 49\\ 0& 1\\ 36& 4 \end{bmatrix}$

$\begin{bmatrix} 45 & -9 \\ -9 & 54 \end{bmatrix}$

$C^{T}C$ is an undefined product.

$\begin{bmatrix} 58 &-7 & 4 \\ -7& 1 & 2 \\ 4 & 2& 40\end{bmatrix}$

$\begin{bmatrix} 9&0 & 36\\49&1 &4 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 58 &-7 & 4 \\ -7& 1 & 2 \\ 4 & 2& 40\end{bmatrix}$

Explanation:

The transpose of a matrix switches the rows and the columns. Therefore, the first column of $C^{T}$ has the same entries, in order, as the first row of , and so forth. Since

$C = \begin{bmatrix} 3&0 & 6\\ -7&1 &2 \end{bmatrix}$ ,

it follows that

$C ^{T} = \begin{bmatrix} 3& -7\\ 0 & 1\\ 6& 2 \end{bmatrix}$

$C^{T}C$

$= \begin{bmatrix} 3& -7\\ 0 & 1\\ 6& 2 \end{bmatrix} \begin{bmatrix} 3&0 & 6\\ -7&1 &2 \end{bmatrix}$

$= \begin{bmatrix} 3(3)+ (-7)(-7)&3(0)+ (-7)(1)& 3(6)+ (-7)(2) \\ 0(3)+1(-7)&0(0)+ 1(1) & 0(6)+1(2) \\ 6(3)+2(-7)&6(0)+ 2 (1) & 6(6)+ 2(2) \end{bmatrix}$

$= \begin{bmatrix} 9+49 &0 + (-7)& 18+ (-14 ) \\ 0 + (-7)&0+1 & 0+2 \\ 18 + (-14)&0+2& 36+4\end{bmatrix}$

$= \begin{bmatrix} 58 &-7 & 4 \\ -7& 1 & 2 \\ 4 & 2& 40\end{bmatrix}$

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Example Question #11 : Matrix Matrix Product

$C = \begin{bmatrix} 3&0 & 6\\ -7&1 &2 \end{bmatrix}$

Evaluate $C^{2}$ .

Possible Answers:

$\begin{bmatrix} 45 & -9 \\ -9 & 54 \end{bmatrix}$

$\begin{bmatrix} 9&0 & 36\\49&1 &4 \end{bmatrix}$

$\begin{bmatrix} 9 & 49\\ 0& 1\\ 36& 4 \end{bmatrix}$

$C^{2}$ is an undefined product.

$\begin{bmatrix} 58 &-7 & 4 \\ -7& 1 & 2 \\ 4 & 2& 40\end{bmatrix}$

Correct answer:

$C^{2}$ is an undefined product.

Explanation:

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. Therefore, for the square of a matrix to be defined, the number of rows and columns in that matrix must be the same; that is, it must be a square matrix. , having two rows and three columns, is not square, so $C^{2}$ cannot exist.

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Linear Algebra : Linear Algebra

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #3 : Matrices

Example Question #4 : Matrices

Example Question #691 : Linear Algebra

Example Question #8 : Matrix Matrix Product

Example Question #2 : Matrix Matrix Product

Example Question #6 : Matrices

Example Question #11 : Matrices

Example Question #11 : Matrices

Example Question #11 : Matrix Matrix Product

Example Question #11 : Matrix Matrix Product

All Linear Algebra Resources

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