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Inverse of a Matrix

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Wize University Linear Algebra Textbook > Matrices

Solving Matrix Equations

6 Activities

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  1. Grab a piece of paper and try this problem yourself.
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e.g. In each case find the matrix A:
 ⁣(a)[2[11−23]−5A‾−1]T=(4A‾T)−1 ⁣;(b)[2A‾T−3I‾]−1=[3111]\!\begin{array}{rr} (a) \left[2\begin{bmatrix} 1&1\\-2&3 \end{bmatrix}-5\underline{A}^{-1}\right]^T=\left( 4\underline{A}^T \right)^{-1}\!;& \hspace{1cm} (b) \left[ 2\underline{A}^T -3\underline{I} \right]^{-1} =\begin{bmatrix} 3&1\\1&1 \end{bmatrix}\end{array}



More Solving Matrix Equations Questions:
Practice: Method of Inverse
Practice Question: Method of Inverse
Given the matrix A=[2130101−12]A=\left[\begin{array}{rrr} 2&1&3\\ 0&1&0\\ 1&-1&2 \end{array}\right],
Practice: Method of Inverse
Practice Question: Method of Inverse
Given the matrix A=[2130101−12]A=\left[\begin{array}{rrr} 2&1&3\\ 0&1&0\\ 1&-1&2 \end{array}\right],
133 - FML 3 - 18.1W e.g. 16
Let   ⁣M‾=[k330(k−1)300(k−2)]\bcb{\boldsymbol{ \ul{M} = \begin{bmatrix} k & 3 & 3 \\ 0 & (k-1) & 3 \\ 0 & 0 & (k-2) \end{bmatrix}}}, for what value(s) of k\bcb{\boldsymbol{ k}} does the system   ⁣M‾x⃗=b⃗\bcb{\boldsymbol{ \ul{M} \vec{x} = \vec{b} }} have a unique solution?
(a)  k=0,  1, 2(b)  k≠0, 1 ,2(c)  k≠0(d)  None of the above\text{(a)}\;\bcb{\boldsymbol{ k = 0, \, \, 1,\, 2}}\qquad\qquad\qquad \text{(b)}\;\bcb{\boldsymbol{ k \ne 0,\, 1\, ,2}}\qquad\qquad\qquad \text{(c)}\;\bcb{\boldsymbol{ k \ne 0}}\qquad\qquad\qquad \text{(d)}\; \text{\bf{None of the above}}
 ~
Practice: Solving Matrix Equations
Solve for AA in the following equation:
A[−122−5]=I2−[2−302]A \left[ \begin{array}{rr} -1&2\\ 2&-5 \end{array} \right] = I_2- \left[ \begin{array}{rr} 2&-3\\ 0&2 \end{array} \right]
Find the inverse of[10221−13−215]and use it to solve the system\text{Find the inverse of}\begin{bmatrix} 1&0&2\\ 2&1&-1\\ 3&-2&15 \end{bmatrix}\text{and use it to solve the system}
𝑥+2𝑧=3𝑥+2𝑧=3
2𝑥+𝑦−𝑧=42𝑥+𝑦−𝑧=4
Practice: Matrix Inverse
Find the inverse of the following matrices:
a.) A=[4534]A=\begin{bmatrix} 4&5\\ 3&4 \end{bmatrix}
b.) B=[−1−2024113−2]B=\begin{bmatrix} -1&-2&0\\ 2&4&1\\ 1&3&-2 \end{bmatrix}
Given the matrix 𝐴=[2130101−12].\text{Given the matrix 𝐴}=\begin{bmatrix} 2&1&3\\ 0&1&0\\ 1&-1&2\\ \end{bmatrix}.
a) find the inverse of 𝐴.
b) solve the system of linear equations 2𝑥+𝑦+3𝑧=32𝑥+𝑦+3𝑧=3
Find the general solution the following system of equations using inverse of the coefficient matrix
2x+y+4z=−3x+3z=−14x+2y+z=2\begin{array}{l}2x+y+4z=-3\\x+3z=-1\\4x+2y+z=2\end{array}
Use the method of inverse to solve the system of linear equations x+2z=32x+y−z=43x−2y+15z=0\begin{array}{c} x+2z&=&3\\ 2x+y-z&=&4\\ 3x-2y+15z&=&0 \end{array}
Use the fact that the matrix
A=[41−420−1−3−14]A=\begin{bmatrix} 4&1&-4\\ 2&0&-1\\ -3&-1&4 \end{bmatrix}
has inverse matrix
Practice: Method of Inverse
Practice Question: Method of Inverse
Given the matrix A=[2130101−12]A=\left[\begin{array}{rrr} 2&1&3\\ 0&1&0\\ 1&-1&2 \end{array}\right],
Practice: Matrix Operations
Find the values of aa and bb such that [a2−11][23]=[10b]T\left[\begin{array}{c} a&2\\ -1&1 \end{array}\right] \left[\begin{array}{c} 2\\ 3 \end{array}\right] = \left[\begin{array}{c} 10&b \end{array}\right]^T
Practice: Matrix Operations
Solve for a, b, ca,\ b,\ c in the following matrix equation:
[1a−12−2130−1][−12b]−[19−1]T=[−8c−2]\begin{bmatrix} 1 &a&-1\\ 2&-2&1\\ 3&0&-1 \end{bmatrix} \begin{bmatrix} -1\\2\\b \end{bmatrix}- \begin{bmatrix} 1&9&-1 \end{bmatrix}^T =\begin{bmatrix} -8\\c\\-2 \end{bmatrix}
Practice: Matrix Inverse
Practice: Solving Matrix Equations
Solve for AA in the following equation:
A[−122−5]=I2−[2−302]A \left[ \begin{array}{rr} -1&2\\ 2&-5 \end{array} \right] = I_2- \left[ \begin{array}{rr} 2&-3\\ 0&2 \end{array} \right]
Practice Question: Using Matrix inverse
Practice Question: Using Matrix Inverse
Let A=[120−1]A=\left[\begin{array}{} 1&2\\0&-1 \end{array}\right] and suppose that 3AB−[1−120]=I23AB- \left[\begin{array}{} 1&-1\\ 2&0 \end{array}\right] =I_2.
Find the matrix BB.
Solving Matrix Equations
Let [3−104]\begin{bmatrix} 3&-1\\ 0&4 \end{bmatrix} be the inverse of the matrix [abcd]\begin{bmatrix} a&b\\c&d \end{bmatrix}.
Find the solution to the following system of linear equations:
ax+by=−2cx+dy=2\begin{array}{rrr} ax+by&=&-2\\ cx+dy&=&2 \end{array}
Solving Matrix Equations
Solve the following system of linear equations using the inverse of the coefficient matrix.
3x+4y=1−2x−y=0\begin{array}{rrr} 3x+4y&=&1\\ -2x-y&=&0 \end{array}
Practice: Matrix Inverse
Solve for AA in the following equation:
A[−122−5]=I2−[2−302]A\begin{bmatrix} -1&2\\ 2&-5 \end{bmatrix} = I_2-\begin{bmatrix} 2&-3\\ 0&2 \end{bmatrix}
Practice: Solving Matrix Equations
Find the matrix CC such that:
(I−2C[24−1−3])−1=[3111]\left(I-2C\left[\begin{array}{rr}2&4\\-1&-3\end{array}\right]\right)^{-1} =\left[\begin{array}{rr}3&1\\1&1\end{array}\right]
Use the fact that the matrix
A=[41−420−1−3−14]A=\begin{bmatrix} 4&1&-4\\ 2&0&-1\\ -3&-1&4 \end{bmatrix}
has inverse matrix
Cramer's Rule
Find yy given the system of linear equations:
[0−1142021−2][xyz]=[214]\left[ \begin{array}{rrr} 0&-1&1\\ 4&2&0\\ 2&1&-2 \end{array} \right] \begin{bmatrix} x\\y\\z\\ \end{bmatrix} = \begin{bmatrix} 2\\1\\4\\ \end{bmatrix}