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Linear Algebra : The Inverse

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Example Question #1 : The Inverse

Calculate $A^{-1}$ , where

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

Possible Answers:

$A^{-1}=\left[\begin{matrix} -\frac{2}{3}& -\frac{5}{6}\\ -\frac{1}{3}& \frac{1}{6} \end{matrix}\right]$

$A^{-1}=\left[\begin{matrix} -\frac{2}{9}& \frac{5}{8}\\ \frac{1}{3}& -\frac{1}{9} \end{matrix}\right]$

$A^{-1}=\left[\begin{matrix} -\frac{2}{3}& \frac{5}{6}\\ \frac{1}{3}& -\frac{1}{6} \end{matrix}\right]$

$A^{-1}=\left[\begin{matrix} -\frac{2}{11}& \frac{5}{11}\\ \frac{1}{11}& -\frac{1}{10} \end{matrix}\right]$

$A^{-1}=\left[\begin{matrix} -\frac{2}{3}& \frac{5}{8}\\ \frac{1}{6}& -\frac{1}{8} \end{matrix}\right]$

Correct answer:

$A^{-1}=\left[\begin{matrix} -\frac{2}{3}& \frac{5}{6}\\ \frac{1}{3}& -\frac{1}{6} \end{matrix}\right]$

Explanation:

The first step, is to create an augmented matrix with the identity Matrix.

$A=\left[ \begin{matrix} 1 & 5 \\ 2 &4 \\ \end{matrix} \right | \left \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right ]$

To find the inverse, all we need to do is get the Identity Matrix on the left hand side.

$2R_1-R_2\rightarrow R_2$

$A=\left[ \begin{matrix} 1 & 5 \\ 0 &6 \\ \end{matrix} \right | \left \begin{matrix} 1 & 0 \\ 2 & -1 \\ \end{matrix} \right ]$

$\frac{1}{6}R_2\rightarrow R_2$

$A=\left[\begin{matrix} 1 & 5\\ 0 & 1 \end{matrix}\right| \left.\begin{matrix} 1& 0\\ \frac{1}{3}& -\frac{1}{6} \end{matrix}\right]$

$-5R_2+R_1\rightarrow R_1$

$A=\left[\begin{matrix} 1 & 0\\ 0 & 1 \end{matrix}\right| \left.\begin{matrix} -\frac{2}{3}& \frac{5}{6}\\ \frac{1}{3}& -\frac{1}{6} \end{matrix}\right]$

Since we have the Identity Matrix on the left hand side, we are done solving for the inverse.

$A^{-1}=\left[\begin{matrix} -\frac{2}{3}& \frac{5}{6}\\ \frac{1}{3}& -\frac{1}{6} \end{matrix}\right]$

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Example Question #2 : The Inverse

Find the inverse of the matrix $\begin{bmatrix} 1 & -2\\ -3& 3\end{bmatrix}$

Possible Answers:

$\begin{bmatrix} -1 & \frac{2}{3}\\ -1& -\frac{1}{3}\end{bmatrix}$

$\begin{bmatrix} 1 & 2\\ -3& 3\end{bmatrix}$

$\begin{bmatrix} -1 & -\frac{2}{3} \\ -1& -\frac{1}{3}\end{bmatrix}$

$\begin{bmatrix} -\frac{3}{8} & \frac{1}{4}\\ -\frac{3}{8}& \frac{1}{8}\end{bmatrix}$

$\begin{bmatrix} -\frac{1}{3} & \frac{2}{3}\\ -1& -1\end{bmatrix}$

Correct answer:

$\begin{bmatrix} -1 & -\frac{2}{3} \\ -1& -\frac{1}{3}\end{bmatrix}$

Explanation:

To find the inverse, first find the determinant. In this case, the determinant is $1\cdot3 - (-3)(-2) = 3 - 6 = -3$

The inverse is found by multiplying $-\frac{1}{3} \begin{bmatrix} 3 & 2\\ 3& 1\end{bmatrix} = \begin{bmatrix} -1 &-\frac{2}{3} \\ -1& -\frac{1}{3}\end{bmatrix}$

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Example Question #3 : The Inverse

Find the inverse of the matrix $\begin{bmatrix} 5 & 0\\ -1& 2\end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 0.2 &0 \\ -0.1&0.5 \end{bmatrix}$

$\begin{bmatrix} 0.2 & 0 \\ 0.1& 0.5\end{bmatrix}$

$\begin{bmatrix} 0.5 & 0\\ -0.1& 0.2 \end{bmatrix}$

$\begin{bmatrix} 50 & 20 \\0 & -10 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 0.2 & 0 \\ 0.1& 0.5\end{bmatrix}$

Explanation:

First, find the determinant: (5)(2) - (0)(-1) = 10 + 0 = 10

Now multiply $\frac{1}{10}$ by the matrix $\begin{bmatrix} 2 &0 \\1 & 5\end{bmatrix}$ , the original matrix with 2 and 5 switched and the signs changed on -1 and 0.

$0.1 \begin{bmatrix} 2&0 \\1 & 5\end{bmatrix} = \begin{bmatrix} 0.2 &0 \\ 0.1 & 0.5\end{bmatrix}$

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Example Question #4 : The Inverse

Find the inverse of matrix A.

$A = \begin{bmatrix} 2&3 \\ 4&-6 \end{bmatrix}$

Possible Answers:

$A^{-1}=\begin{bmatrix} 2&-6 \\ 4&3 \end{bmatrix}$

$A^{-1} = \begin{bmatrix} 2& 4\\ 3&-6 \end{bmatrix}$

Matrix A is not invertible.

$A^{-1}=\begin{bmatrix} -6&4 \\3 & 2\end{bmatrix}$

$A^{-1}=\begin{bmatrix} -6&2 \\ -3&-4 \end{bmatrix}$

Correct answer:

Matrix A is not invertible.

Explanation:

For any 2x2 matrix, to determine if it is invertible, we must first calculate its determinant. If the determinant is equal to 0, then the matrix is not invertible. If it isn’t equal to 0, then its inverse can be found using this formula:

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

$A^{-1}= \frac{1}{ad-bc}\begin{bmatrix} d& -b\\ -c& a\end{bmatrix}$

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Example Question #5 : The Inverse

Find the inverse of matrix A.

$A=\begin{bmatrix} 4&3 \\ 6&5 \end{bmatrix}$

Possible Answers:

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

$A^{-1}=\begin{bmatrix} 5&-3 \\ -6&4 \end{bmatrix}$

Matrix A is not invertible.

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

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

Correct answer:

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

Explanation:

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

$A^{-1}= \frac{1}{ad-bc}\begin{bmatrix} d& -b\\ -c& a\end{bmatrix}$

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Example Question #1 : The Inverse

Determine the inverse of matrix A where

$A=\begin{bmatrix} 32&1 \\ 5&8 \\ -1&6 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 5&32 &-1 \\ 1&6 &1 \end{bmatrix}$

$\begin{bmatrix} 1/32&5/6 \\ 8/9& 21/23\\ 1/2& 0 \end{bmatrix}$

$\begin{bmatrix} 32&5 &-1 \\ 1&8 &6 \end{bmatrix}$

Not Possible

Correct answer:

Not Possible

Explanation:

The matrix is not square so it does not have an inverse.

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Example Question #2 : The Inverse

Determine the inverse of matrix A where

$A = \begin{bmatrix} 1&5 \\ 4&20 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 20&-5 \\ -4& 1 \end{bmatrix}$

$\begin{bmatrix} 1/5&1 \\ 4/5&4 \end{bmatrix}$

Inverse does not exist

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

Correct answer:

Inverse does not exist

Explanation:

The matrix is square, so it could have an inverse. Next we find the determinant. This matrix has a determinant of 0, so it does not have an inverse.

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Example Question #1 : The Inverse

Determine the inverse of matrix A where

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

Possible Answers:

$\begin{bmatrix} -5/9&1/9 \\ 2/9&2/9 \end{bmatrix}$

$\begin{bmatrix} 2/9&1/9 \\ -5/9&2/9 \end{bmatrix}$

$\begin{bmatrix} 1/9&2/9 \\ 2/9&-5/9 \end{bmatrix}$

$\begin{bmatrix} 2/9&-5/9 \\ 1/9&2/9 \end{bmatrix}$

Inverse does not exist.

Correct answer:

$\begin{bmatrix} 2/9&1/9 \\ -5/9&2/9 \end{bmatrix}$

Explanation:

To find the inverse of a matrix first look to verify that the matrix is square. If it is not square, it does not have an inverse. Next, you must find the determinant. If the determinant is 0, then the matrix does not have an inverse. The determinant for this matrix is ad-bc = 9, therefore it has an inverse. To find the inverse of a 2x2 matrix we first write it in augmented form.

$\begin{bmatrix} 2&-1 &1 &0 \\ 5&2 &0 & 1 \end{bmatrix}$ First we will divide R1/2 $\begin{bmatrix} 1&-1/2 &1/2 &0 \\ 5&2 &0& 1\end{bmatrix}$ next we will eliminate the first column by taking R2-5R1 $\begin{bmatrix} 1&-1/2 &1/2 &0 \\ 0&9/2 &-5/2& 1 \end{bmatrix}$ , next we will divide 9R2/2 to set the second pivot. $\begin{bmatrix} 1&-1/2 &1/2 &0 \\ 0&1 &-5/9& 2/9 \end{bmatrix}$ . Next we will eliminate the second column by taking R1+1/2R2. $\begin{bmatrix} 1&0 &2/9 &1/9 \\ 0&1 &-5/9& 2/9 \end{bmatrix}$ . Now that we have the identity matrix on the left, our answer is on the right. There is, however, an easier way to determine the inverse of a 2x2 matrix. The trick is to swap the numbers in spots a and d, put negatives in front of the numbers in spots b and c and then divide everything by the determinant. For this example, $\begin{bmatrix} 2&1 \\ -5& 2 \end{bmatrix}$ then divide by the determinant which is 9 and simplify.

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Example Question #3 : The Inverse

Determine the inverse of matrix A where

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

Possible Answers:

The inverse does not exist.

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

$A = \begin{bmatrix} 2&3 & 7\\ 0&7&4 \end{bmatrix}$

$A = \begin{bmatrix} 3/3&1 & 3\\ 7/2&0 &1/2 \end{bmatrix}$

Correct answer:

The inverse does not exist.

Explanation:

The matrix is not square, so it does not have an inverse.

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Example Question #4 : The Inverse

Determine the inverse of matrix A where

$A=\begin{bmatrix} 2& 3\\ 4& 6 \end{bmatrix}$

Possible Answers:

This matrix does not have an inverse.

$A=\begin{bmatrix} 3& 2\\ 6& 4 \end{bmatrix}$

$A=\begin{bmatrix} 2/3& 1\\ 4/3& 2 \end{bmatrix}$

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

Correct answer:

This matrix does not have an inverse.

Explanation:

To determine the inverse of a matrix, you must first verify that the matrix is square. Next calculate the determinant. The determinant for this matrix is 0, so it does not have an inverse.

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Linear Algebra : The Inverse

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #1 : The Inverse

Example Question #2 : The Inverse

Example Question #3 : The Inverse

Example Question #4 : The Inverse

Example Question #5 : The Inverse

Example Question #1 : The Inverse

Example Question #2 : The Inverse

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