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Linear Algebra : Operations and Properties

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Example Questions

Example Question #261 : Operations And Properties

A square matrix is invertible if and only if-

Possible Answers:

it is similar to the zero matrix

its determinant is .

None of the other answers

it is diagonalizable

Correct answer:

None of the other answers

Explanation:

All of these are criteria for a matrix NOT to be invertible, (except for diagonalization; some diagonalizable matrices are invertible and some aren't.)

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Example Question #35 : The Inverse

$A = \begin{bmatrix} \frac{1}{2} & \frac{1}{3} \\ \\ \frac{2}{3} & \frac{1}{6}\end{bmatrix}$

Find $A^{-1}$ .

Possible Answers:

$\begin{bmatrix}-\frac{18}{ 5} &\frac{12}{ 5} \\ \\ \frac{24}{ 5} & -\frac{6}{ 5} \end{bmatrix}$

$\begin{bmatrix}-\frac{6}{ 5} &\frac{12}{ 5} \\ \\ \frac{24}{ 5} & -\frac{18}{ 5} \end{bmatrix}$

$\begin{bmatrix}\frac{6}{ 5} &\frac{12}{ 5} \\ \\\frac{24}{ 5} & \frac{18}{ 5} \end{bmatrix}$

$\begin{bmatrix}\frac{18}{ 5} &\frac{12}{ 5} \\ \\ \frac{24}{ 5} & \frac{6}{ 5} \end{bmatrix}$

$\begin{bmatrix}-\frac{6}{ 5} &-\frac{12}{ 5} \\ \\- \frac{24}{ 5} & -\frac{18}{ 5} \end{bmatrix}$

Correct answer:

$\begin{bmatrix}-\frac{6}{ 5} &\frac{12}{ 5} \\ \\ \frac{24}{ 5} & -\frac{18}{ 5} \end{bmatrix}$

Explanation:

The inverse of a two-by-two matrix

$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21}& a_{22} \end{bmatrix}$

is the matrix

$A^{-1} =\frac{1}{D} \begin{bmatrix} a_{22} & -a_{12} \\ -a_{21}& a_{11} \end{bmatrix}$ .

is the determinant of the matrix, which is the product of the main diagonal elements minus the product of the other two:

$D = a_{11}a_{22}- a_{12}a_{21}$

$= \frac{1}{2} \left ( \frac{1}{6} \right ) - \frac{1}{3}\left ( \frac{2}{3} \right )$

$= \frac{1}{12} - \frac{2}{9}$

$= -\frac{5}{36}$

Therefore,

$A^{-1} =\frac{1}{ -\frac{5}{36}} \begin{bmatrix} \frac{1}{6} & -\frac{1}{3} \\ \\ -\frac{2}{3} & \frac{1}{2}\end{bmatrix}$

$=-\frac{36}{ 5} \begin{bmatrix} \frac{1}{6} & -\frac{1}{3} \\ \\ -\frac{2}{3} & \frac{1}{2}\end{bmatrix}$

$= \begin{bmatrix}-\frac{36}{ 5} \left ( \frac{1}{6} \right ) &-\frac{36}{ 5} \left ( -\frac{1}{3} \right ) \\ \\ -\frac{36}{ 5} \left ( -\frac{2}{3} \right ) & -\frac{36}{ 5} \left ( \frac{1}{2} \right ) \end{bmatrix}$

$= \begin{bmatrix}-\frac{6}{ 5} &\frac{12}{ 5} \\ \\ \frac{24}{ 5} & -\frac{18}{ 5} \end{bmatrix}$

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Example Question #41 : The Inverse

$A= \begin{bmatrix} 0. 3 & 1. 7 \\ 0.5 & 2.3 \end{bmatrix}$

Find $A^{-1}$ .

Possible Answers:

$\begin{bmatrix} 14.375 &-10.625 \\- 3.125 & 1.875 \end{bmatrix}$

$\begin{bmatrix} -14.375 &10.625 \\ 3.125 &-1.875 \end{bmatrix}$

$\begin{bmatrix} -1.875 &10.625 \\ 3.125 & -14.375 \end{bmatrix}$

$\begin{bmatrix} -1.875 &-10.625 \\- 3.125 & -14.375 \end{bmatrix}$

$\begin{bmatrix} -14.375 &-10.625 \\- 3.125 &-1.875 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} -14.375 &10.625 \\ 3.125 &-1.875 \end{bmatrix}$

Explanation:

The inverse of a two-by-two matrix

$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21}& a_{22} \end{bmatrix}$

is the matrix

$A^{-1} =\frac{1}{D} \begin{bmatrix} a_{22} & -a_{12} \\ -a_{21}& a_{11} \end{bmatrix}$ .

is the determinant of the matrix, which is the product of the main diagonal elements minus the product of the other two:

$D = a_{11}a_{22}- a_{12}a_{21}$

= 0.3(2.3)- 1.7(0.5)

= 0.69 - 0.85

= -0.16

Therefore,

$A^{-1}=\frac{1}{-0.16} \begin{bmatrix} 2. 3 &- 1. 7 \\ -0.5 & 0.3 \end{bmatrix}$

$=\begin{bmatrix} \frac{1}{-0.16} \cdot 2. 3 & \frac{1}{-0.16} \cdot(- 1. 7) \\ \\ \frac{1}{-0.16} \cdot(-0.5) & \frac{1}{-0.16} \cdot0.3 \end{bmatrix}$

$=\begin{bmatrix} -14.375 &10.625 \\ 3.125 &-1.875 \end{bmatrix}$

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Example Question #1 : Orthogonal Matrices

Determine if the following matrix is orthogonal or not.

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

Possible Answers:

is an orthogonal matrix

is not an orthogonal matrix

Correct answer:

is an orthogonal matrix

Explanation:

To determine if a matrix is orthogonal, we need to multiply the matrix by it's transpose, and see if we get the identity matrix.

$A=\begin{bmatrix} -1& 0\\ 0& 1 \end{bmatrix}$ , $A^T=\begin{bmatrix} -1& 0\\ 0& 1 \end{bmatrix}$

$A.A^T=\begin{bmatrix} (-1)(-1)&(0)(0) \\ (0)(0)& (1)(1) \end{bmatrix}=\begin{bmatrix} 1&0\\ 0& 1 \end{bmatrix}$

Since we get the identity matrix, then we know that is an orthogonal matrix.

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Example Question #1 : Orthogonal Matrices

$\begin{align*}&\text{Conclude whether or not the matrix }A=\begin{bmatrix}-0.60587&0.79556\\0.79556&0.60587\end{bmatrix}\\&\text{is orthogonal.}\end{align*}$

Possible Answers:

$\begin{align*}&\text{The matrix is orthogonal.}\end{align*}$

$\begin{align*}&\text{The matrix is not orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}&\text{The matrix is orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.60587&0.79556\\0.79556&0.60587\end{bmatrix}= \\&\text{The matrix is orthogonal.}\end{align*}$

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Example Question #3 : Orthogonal Matrices

$\begin{align*}&\text{Conclude whether or not the matrix }A=\begin{bmatrix}0.68567&0.12975&-0.71626\\0.14807&0.93855&0.31176\\0.71269&-0.31982&0.62433\end{bmatrix}\\&\text{is orthogonal.}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}0.68567&0.12975&-0.71626\\0.14807&0.93855&0.31176\\0.71269&-0.31982&0.62433\end{bmatrix}=1\\&\text{The matrix is orthogonal.}\end{align*}$

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Example Question #4 : Orthogonal Matrices

$\begin{align*}&\text{Is }A=\begin{bmatrix}-0.85616&0.46933\\0.46933&0.96236\end{bmatrix}\\&\text{orthogonal?}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.85616&0.46933\\0.46933&0.96236\end{bmatrix}=-1.0442\\&\text{The matrix is not orthogonal.}\end{align*}$

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Example Question #1 : Orthogonal Matrices

$\begin{align*}&\text{Conclude whether or not the matrix }A=\begin{bmatrix}-0.57605&-0.81742\\-0.81742&0.57605\end{bmatrix}\\&\text{is orthogonal.}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.57605&-0.81742\\-0.81742&0.57605\end{bmatrix}=-1\\&\text{The matrix is orthogonal.}\end{align*}$

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Example Question #6 : Orthogonal Matrices

$\begin{align*}&\text{Is the matrix }A=\begin{bmatrix}-0.44125&-0.89739\\-0.89739&0.44125\end{bmatrix}\\&\text{orthogonal?}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.44125&-0.89739\\-0.89739&0.44125\end{bmatrix}=-1\\&\text{The matrix is orthogonal.}\end{align*}$

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Example Question #7 : Orthogonal Matrices

$\begin{align*}&\text{Is the matrix }A=\begin{bmatrix}-0.62126&-0.0069882&0.77059\\-0.86406&-0.14727&-0.61555\\0.011558&-1.0415&0.043005\end{bmatrix}\\&\text{orthogonal?}\end{align*}$

Possible Answers:

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

$\begin{align*}\text{The matrix is orthogonal.}\end{align*}$

Correct answer:

$\begin{align*}\text{The matrix is not orthogonal.}\end{align*}$

Explanation:

$\begin{align*}&\text{There are multiple ways to determine whether or not a matrix is orthogonal.}\\&\text{For instance, a matrix is orthogonal if its transpose equals its inverse:}\\&A^{-1}=A^{T}\\&\text{Another method is to see if the product of it and its tranpose is the identity matrix:}\\&A^{T}A=AA^{T}=I\\&\text{If so, it is orthgonal.}\\&\text{Finally, if the matrix's determinant is }\pm1\\&\text{then it is orthgonal. Using this last method:}\\&det\begin{bmatrix}-0.62126&-0.0069882&0.77059\\-0.86406&-0.14727&-0.61555\\0.011558&-1.0415&0.043005\end{bmatrix}=1.0968\\&\text{The matrix is not orthogonal.}\end{align*}$

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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 #261 : Operations And Properties

Example Question #35 : The Inverse

Example Question #41 : The Inverse

Example Question #1 : Orthogonal Matrices

Example Question #1 : Orthogonal Matrices

Example Question #3 : Orthogonal Matrices

Example Question #4 : Orthogonal Matrices

Example Question #1 : Orthogonal Matrices

Example Question #6 : Orthogonal Matrices

Example Question #7 : Orthogonal Matrices

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