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Linear Algebra : Operations and Properties

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Example Questions

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Example Question #11 : The Transpose

$\begin{align*}&\text{Find the transpose of }\begin{bmatrix}-4&-4&13&-8&13\\-6&-17&18&12&3\\1&-9&-6&-7&18\\18&-17&-18&-7&6\end{bmatrix}^{T}\end{align*}$

Possible Answers:

$\begin{bmatrix}-4&-4&13&-8&13\\-6&-17&18&12&3\\1&-9&-6&-7&18\\18&-17&-18&-7&6\end{bmatrix}$

$\begin{bmatrix}18&1&-6&-4\\-17&-9&-17&-4\\-18&-6&18&13\\-7&-7&12&-8\\6&18&3&13\end{bmatrix}$

$\begin{bmatrix}-4&-6&1&18\\-4&-17&-9&-17\\13&18&-6&-18\\-8&12&-7&-7\\13&3&18&6\end{bmatrix}$

$\begin{bmatrix}6&18&3&13\\-7&-7&12&-8\\-18&-6&18&13\\-17&-9&-17&-4\\18&1&-6&-4\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-4&-4&13&-8&13\\-6&-17&18&12&3\\1&-9&-6&-7&18\\18&-17&-18&-7&6\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Note the superscript, T, in our problem statement; this denotes the transpose.}\\&\text{The transpose of the transpose of a matrix is the orignal matrix, so:}\\&\begin{bmatrix}-4&-4&13&-8&13\\-6&-17&18&12&3\\1&-9&-6&-7&18\\18&-17&-18&-7&6\end{bmatrix}^{TT}=\begin{bmatrix}-4&-4&13&-8&13\\-6&-17&18&12&3\\1&-9&-6&-7&18\\18&-17&-18&-7&6\end{bmatrix}\end{align*}$

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Example Question #11 : The Transpose

$\begin{align*}&\text{Find }A^{T}\text{ for }A=\begin{bmatrix}-11&14\\-18&7\\7&-6\\-17&-9\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}14&7&-6&-9\\-11&-18&7&-17\end{bmatrix}$

$\begin{bmatrix}-17&7&-18&-11\\-9&-6&7&14\end{bmatrix}$

$\begin{bmatrix}-9&-6&7&14\\-17&7&-18&-11\end{bmatrix}$

$\begin{bmatrix}-11&-18&7&-17\\14&7&-6&-9\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-11&-18&7&-17\\14&7&-6&-9\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Finding the transpose of a matrix is simply a matter of making}\\&\text{the first row the first column, the second row the second column, etc:}\\&A=\begin{bmatrix}-11&14\\-18&7\\7&-6\\-17&-9\end{bmatrix}\\&A^{T}=\begin{bmatrix}-11&-18&7&-17\\14&7&-6&-9\end{bmatrix}\end{align*}$

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Example Question #11 : The Transpose

$\begin{align*}&\text{Find the transpose of }A=\begin{bmatrix}14&-3&18&17&-12\\3&2&-16&20&-11\\-8&-14&-15&17&16\\15&-10&3&14&-8\\-17&-7&15&4&-3\\8&-10&17&12&20\\-14&14&-7&-9&-10\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}-14&8&-17&15&-8&3&14\\14&-10&-7&-10&-14&2&-3\\-7&17&15&3&-15&-16&18\\-9&12&4&14&17&20&17\\-10&20&-3&-8&16&-11&-12\end{bmatrix}$

$\begin{bmatrix}14&3&-8&15&-17&8&-14\\-3&2&-14&-10&-7&-10&14\\18&-16&-15&3&15&17&-7\\17&20&17&14&4&12&-9\\-12&-11&16&-8&-3&20&-10\end{bmatrix}$

$\begin{bmatrix}-12&-11&16&-8&-3&20&-10\\17&20&17&14&4&12&-9\\18&-16&-15&3&15&17&-7\\-3&2&-14&-10&-7&-10&14\\14&3&-8&15&-17&8&-14\end{bmatrix}$

$\begin{bmatrix}-10&20&-3&-8&16&-11&-12\\-9&12&4&14&17&20&17\\-7&17&15&3&-15&-16&18\\14&-10&-7&-10&-14&2&-3\\-14&8&-17&15&-8&3&14\end{bmatrix}$

Correct answer:

$\begin{bmatrix}14&3&-8&15&-17&8&-14\\-3&2&-14&-10&-7&-10&14\\18&-16&-15&3&15&17&-7\\17&20&17&14&4&12&-9\\-12&-11&16&-8&-3&20&-10\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Determining the transpose of a matrix is relatively simple,}\\&\text{though attention to detail should be exercised.}\\&\text{In creating the transpose, the first row becomes the first column,}\\&\text{the second row the second column, etc:}\\&A=\begin{bmatrix}14&-3&18&17&-12\\3&2&-16&20&-11\\-8&-14&-15&17&16\\15&-10&3&14&-8\\-17&-7&15&4&-3\\8&-10&17&12&20\\-14&14&-7&-9&-10\end{bmatrix}\\&A^{T}=\begin{bmatrix}14&3&-8&15&-17&8&-14\\-3&2&-14&-10&-7&-10&14\\18&-16&-15&3&15&17&-7\\17&20&17&14&4&12&-9\\-12&-11&16&-8&-3&20&-10\end{bmatrix}\end{align*}$

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Example Question #21 : The Transpose

$\begin{align*}&\text{Find }\begin{bmatrix}-10&-6&-9\\3&-13&-2\\11&4&-20\end{bmatrix}^{T}\end{align*}$

Possible Answers:

$\begin{bmatrix}-10&3&11\\-6&-13&4\\-9&-2&-20\end{bmatrix}$

$\begin{bmatrix}-20&-2&-9\\4&-13&-6\\11&3&-10\end{bmatrix}$

$\begin{bmatrix}11&3&-10\\4&-13&-6\\-20&-2&-9\end{bmatrix}$

$\begin{bmatrix}-9&-2&-20\\-6&-13&4\\-10&3&11\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-10&3&11\\-6&-13&4\\-9&-2&-20\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Superscript T denotes the transpose of a matrix. Finding this transpose requies only}\\&\text{treating the first row as the first column, the second as row the second column, etc:}\\&\begin{bmatrix}-10&-6&-9\\3&-13&-2\\11&4&-20\end{bmatrix}^{T}=\begin{bmatrix}-10&3&11\\-6&-13&4\\-9&-2&-20\end{bmatrix}\end{align*}$

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Example Question #22 : The Transpose

$\begin{align*}&\text{Find the transpose of }A=\begin{bmatrix}-8&15&-1&-13&5\\3&10&6&10&-1\\-9&3&6&16&14\\2&5&-15&9&-14\end{bmatrix}\end{align*}$

Possible Answers:

$\begin{bmatrix}2&-9&3&-8\\5&3&10&15\\-15&6&6&-1\\9&16&10&-13\\-14&14&-1&5\end{bmatrix}$

$\begin{bmatrix}5&-1&14&-14\\-13&10&16&9\\-1&6&6&-15\\15&10&3&5\\-8&3&-9&2\end{bmatrix}$

$\begin{bmatrix}-14&14&-1&5\\9&16&10&-13\\-15&6&6&-1\\5&3&10&15\\2&-9&3&-8\end{bmatrix}$

$\begin{bmatrix}-8&3&-9&2\\15&10&3&5\\-1&6&6&-15\\-13&10&16&9\\5&-1&14&-14\end{bmatrix}$

Correct answer:

$\begin{bmatrix}-8&3&-9&2\\15&10&3&5\\-1&6&6&-15\\-13&10&16&9\\5&-1&14&-14\end{bmatrix}$

Explanation:

$\begin{align*}&\text{Determining the transpose of a matrix is relatively simple,}\\&\text{though attention to detail should be exercised.}\\&\text{In creating the transpose, the first row becomes the first column,}\\&\text{the second row the second column, etc:}\\&A=\begin{bmatrix}-8&15&-1&-13&5\\3&10&6&10&-1\\-9&3&6&16&14\\2&5&-15&9&-14\end{bmatrix}\\&A^{T}=\begin{bmatrix}-8&3&-9&2\\15&10&3&5\\-1&6&6&-15\\-13&10&16&9\\5&-1&14&-14\end{bmatrix}\end{align*}$

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Example Question #23 : The Transpose

True or false: The transpose of a matrix with six rows and seven columns has seven rows and six columns.

Possible Answers:

False

True

Correct answer:

True

Explanation:

The transposition of a matrix switches the rows and the columns, so the number of rows in the original matrix is equal to the number of columns in the transpose, and vice versa. Therefore, a matrix with six rows and seven columns has seven rows and six columns.

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Example Question #24 : The Transpose

$A = \begin{bmatrix} 3 & 0 & -3 \\ 6& 0 & 9 \end{bmatrix}$

Which of the following is equal to $A^{T}$ ?

Possible Answers:

$A^{T} = \begin{bmatrix} 6& 0 & 9 \\ 3 & 0 & -3 \end{bmatrix}$

$A^{T} = \begin{bmatrix} -3 & 0 & 3 \\ -6& 0 &- 9 \end{bmatrix}$

$A ^{T}= \begin{bmatrix}6 & 3 \\ 0 & 0 \\ -9 & 3 \end{bmatrix}$

The transpose of does not exist.

$A ^{T}= \begin{bmatrix} 3 &6 \\ 0 & 0 \\ -3 & 9 \end{bmatrix}$

Correct answer:

$A ^{T}= \begin{bmatrix} 3 &6 \\ 0 & 0 \\ -3 & 9 \end{bmatrix}$

Explanation:

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

$A = \begin{bmatrix} 3 & 0 & -3 \\ 6& 0 & 9 \end{bmatrix}$

it follows that

$A ^{T}= \begin{bmatrix} 3 &6 \\ 0 & 0 \\ -3 & 9 \end{bmatrix}$

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Example Question #25 : The Transpose

Let and be any two matrices of the same dimensions.

True or false:

It must hold that $(A+B)^{T} = A^{T}+ B^{T}$ .

Possible Answers:

False

True

Correct answer:

True

Explanation:

The transpose of the sum of two matrices is indeed equal to the sum of their transposes. Let us look at the two-by-two case - this reasoning can be generalized.

$A = \begin{bmatrix} a& b\\ c& d \end{bmatrix}$ and $B = \begin{bmatrix} w & x\\ y & z \end{bmatrix}$

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

$A^{T} = \begin{bmatrix} a& c\\ b& d \end{bmatrix}$ and $B ^{T}= \begin{bmatrix} w & y\\x& z \end{bmatrix}$ .

Matrices are added term by term, so

$A^{T}+ B ^{T}$

$= \begin{bmatrix} a& c\\ b& d \end{bmatrix} + \begin{bmatrix} w & y\\x& z \end{bmatrix}$

$= \begin{bmatrix} a+w & c+y\\ b+x& d+z \end{bmatrix}$

$(A+B)^{T}$ , the transpose of the sum of the original matrices, can be found by first adding the matrices termwise:

A + B

$= \begin{bmatrix} a& b\\ c& d \end{bmatrix}+ \begin{bmatrix} w & x\\ y & z \end{bmatrix}$

$= \begin{bmatrix} a+w &b+x \\ c+y & d+z \end{bmatrix}$

Take the transpose of the sum:

$(A+B)^{T}= \begin{bmatrix} a+w & c+y\\ b+x& d+z \end{bmatrix}$

Indeed, $(A+B)^{T} = A^{T}+ B^{T}$ .

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Example Question #21 : The Transpose

is a three-by-three matrix with determinant 0.

True, false, or undetermined: does not have a transpose.

Possible Answers:

False

True

Undetermined

Correct answer:

False

Explanation:

Every matrix has a transpose regardless of the value of its determinant; finding the transpose is a matter of repositioning the elements - a concept independent of determinant - so that its rows are the columns of the transpose, and vice versa.

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Example Question #21 : The Transpose

is a two-by-two matrix such that

$A + A^{T} = \begin{bmatrix} -10& 0\\ 0& 16 \end{bmatrix}$

Which of the following could be?

Possible Answers:

$A = \begin{bmatrix} -5 &12 \\ 12& 8 \end{bmatrix}$

$A = \begin{bmatrix} -10 &- 7\\ -7& 16 \end{bmatrix}$

$A = \begin{bmatrix} -5 &- 11 \\ 11 & 8 \end{bmatrix}$

$A = \begin{bmatrix} -5 & - 7 \\ -7& 8 \end{bmatrix}$

$A = \begin{bmatrix} -10 &- 11 \\ 11 & 16 \end{bmatrix}$

Correct answer:

$A = \begin{bmatrix} -5 &- 11 \\ 11 & 8 \end{bmatrix}$

Explanation:

Since is a two-by-two matrix, we will let

$A = \begin{bmatrix} a& b\\ c& d \end{bmatrix}$ ,

where , , , and represent real values.

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

$A ^{T}= \begin{bmatrix} a&c\\ b& d \end{bmatrix}$

and

$A+A^{T} = \begin{bmatrix} a& b\\ c& d \end{bmatrix} + \begin{bmatrix} a&c\\ b& d \end{bmatrix}$

Add two matrices termwise:

$A+A^{T} = \begin{bmatrix} a+a & b+c\\ c+b& d+d \end{bmatrix}$

$= \begin{bmatrix} 2a & b+c\\ c+b& 2d \end{bmatrix}$

Since

$A + A^{T} = \begin{bmatrix} -10& 0\\ 0& 16 \end{bmatrix}$

or, equivalently,

$\begin{bmatrix} 2a & b+c\\ c+b& 2d \end{bmatrix}= \begin{bmatrix} -10& 0\\ 0& 16 \end{bmatrix}$

$\begin{bmatrix} 2a & b+c\\ b+c& 2d \end{bmatrix}= \begin{bmatrix} -10& 0\\ 0& 16 \end{bmatrix}$

The following must hold:

2a= -10 ,

a= -5

2d = 16 ,

d = 8

and

b+ c = 0 ,

or, equivalently,

c = -b .

Therefore, any matrix of the form

$A = \begin{bmatrix} -5 & b\\ -b & 8 \end{bmatrix}$

for some real would make the statement correct. The only choice matching this pattern is the matrix

$A = \begin{bmatrix} -5 &- 11 \\ 11 & 8 \end{bmatrix}$ .

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Linear Algebra : Operations and Properties

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #11 : The Transpose

Example Question #11 : The Transpose

Example Question #11 : The Transpose

Example Question #21 : The Transpose

Example Question #22 : The Transpose

Example Question #23 : The Transpose

Example Question #24 : The Transpose

Example Question #25 : The Transpose

Example Question #21 : The Transpose

Example Question #21 : The Transpose

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