Practice: Matrix Inverse

Solving Matrix Equations

6 Activities

Wize University Linear Algebra Textbook > Matrices

Matrix Inverse Algorithm

4 Activities

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Find the inverse of the following matrices:
a.) A=[4534]A=\begin{bmatrix}
4&5\\
3&4
\end{bmatrix}A=[43​54​]


b.) B=[−1−2024113−2]B=\begin{bmatrix}
-1&-2&0\\
2&4&1\\
1&3&-2
\end{bmatrix}B=​−121​−243​01−2​​


2. Find all values of k for which the matrix [2−6−1k2−22]\begin{bmatrix}
2&-6\\
-1&k^2-22
\end{bmatrix}[2−1​−6k2−22​] has no inverse.


3. Find all values of k so that the matrix [−2003102−1k]\begin{bmatrix}
-2&0&0\\
3&1&0\\
2&-1&k
\end{bmatrix}​−232​01−1​00k​​ has an inverse.


4. Let [3−104]\begin{bmatrix}
3&-1\\
0&4
\end{bmatrix}[30​−14​] be the inverse of the matrix [abcd]\begin{bmatrix}
a&b\\c&d
\end{bmatrix}[ac​bd​] . Find the solution to the following system of linear equations:
ax+by=−2cx+dy=2\begin{array}{}
ax+by=-2\\
cx+dy=2
\end{array}ax+by=−2cx+dy=2​


5. Solve the following system of linear equations using the method of inverse.
3x+4y=1−2x−y=0\begin{array}{}
3x+4y=1\\
-2x-y=0
\end{array}3x+4y=1−2x−y=0​


6. If A=[abcdefghi]A=
\begin{bmatrix}
a&b&c\\
d&e&f\\
g&h&i
\end{bmatrix}A=​adg​beh​cfi​​ is an invertible matrix and C=AA−1C=AA^{-1}C=AA−1, find the first row of CCC .


7. Solve for AAA 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}A[−12​2−5​]=I2​−[20​−32​]

I have answered this question

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]A=​201​11−1​302​​,

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]A=​201​11−1​302​​,

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}}}M​=​k00​3(k−1)0​33(k−2)​​, for what value(s) of k\bcb{\boldsymbol{ k}}k does the system   ⁣M‾x⃗=b⃗\bcb{\boldsymbol{ \ul{M} \vec{x} = \vec{b} }}M​x=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}}(a)k=0,1,2(b)k=0,1,2(c)k=0(d)None of the above
 ~ 

Practice: Solving Matrix Equations
Solve for AAA 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]A[−12​2−5​]=I2​−[20​−32​]

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}Find the inverse of​123​01−2​2−115​​and use it to solve the system
𝑥+2𝑧=3𝑥+2𝑧=3x+2z=3
2𝑥+𝑦−𝑧=42𝑥+𝑦−𝑧=42x+y−z=4

Given the matrix 𝐴=[2130101−12].\text{Given the matrix 𝐴}=\begin{bmatrix}
2&1&3\\
0&1&0\\
1&-1&2\\
\end{bmatrix}.Given the matrix A=​201​11−1​302​​.
a) find the inverse of 𝐴. 
b) solve the system of linear equations          2𝑥+𝑦+3𝑧=32𝑥+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}2x+y+4z=−3x+3z=−14x+2y+z=2​

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}x+2z2x+y−z3x−2y+15z​===​340​

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}A=​42−3​10−1​−4−14​​  
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]A=​201​11−1​302​​,

Practice: Matrix Operations

Find the values of aaa and bbb 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[a−1​21​][23​]=[10​b​]T

Practice: Matrix Operations

Solve for a, b, ca,\ 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}​123​a−20​−11−1​​​−12b​​−[1​9​−1​]T=​−8c−2​​

Practice: Matrix Inverse

Practice: Solving Matrix Equations
Solve for AAA 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]A[−12​2−5​]=I2​−[20​−32​]

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]A=[10​2−1​] and suppose that 3AB−[1−120]=I23AB-

\left[\begin{array}{}
1&-1\\
2&0
\end{array}\right]

=I_23AB−[12​−10​]=I2​. 
Find the matrix BBB.

Solving Matrix Equations

Let [3−104]\begin{bmatrix}
3&-1\\
0&4
\end{bmatrix}[30​−14​] be the inverse of the matrix [abcd]\begin{bmatrix}
a&b\\c&d
\end{bmatrix}[ac​bd​].
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}ax+bycx+dy​==​−22​

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}3x+4y−2x−y​==​10​

Practice: Matrix Inverse

Solve for AAA 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}A[−12​2−5​]=I2​−[20​−32​]

Inverse of a Matrix

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}(a)[2[1−2​13​]−5A​−1]T=(4A​T)−1;​(b)[2A​T−3I​]−1=[31​11​]​

Practice: Solving Matrix Equations

Find the matrix CCC 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](I−2C[2−1​4−3​])−1=[31​11​]

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}A=​42−3​10−1​−4−14​​
has inverse matrix

Cramer's Rule

Find yyy 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}​042​−121​10−2​​​xyz​​=​214​​

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]A=​201​11−1​302​​,

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]A=​201​11−1​302​​,

Practice: Matrix Inverse

Practice Question: Matrix Inverse
For what value(s) of k will the matrix[1210150k1]\begin{bmatrix}
1&2&1\\
0&1&5\\
0&k&1
\end{bmatrix}​100​21k​151​​ be invertible?

Practice: Matrix Inverse

Practice Question: Matrix Inverse
For what value(s) of k will the matrix[1210150k1]\begin{bmatrix}
1&2&1\\
0&1&5\\
0&k&1
\end{bmatrix}​100​21k​151​​ be invertible?

Practice: Matrix Inverse

Practice Question: Matrix Inverse
For what value(s) of k will the matrix[1210150k1]\begin{bmatrix}
1&2&1\\
0&1&5\\
0&k&1
\end{bmatrix}​100​21k​151​​ be invertible?

Practice: Matrix Inverse

Practice Question: Matrix Inverse
For what value(s) of k will the matrix[1210150k1]\begin{bmatrix}
1&2&1\\
0&1&5\\
0&k&1
\end{bmatrix}​100​21k​151​​ be invertible?

Practice: Matrix Inverse

Practice Question: Matrix Inverse
For what value(s) of k will the matrix[1210150k1]\begin{bmatrix}
1&2&1\\
0&1&5\\
0&k&1
\end{bmatrix}​100​21k​151​​ be invertible?

Practice: Matrix Inverse

Practice Question: Matrix Inverse
For what value(s) of k will the matrix[1210150k1]\begin{bmatrix}
1&2&1\\
0&1&5\\
0&k&1
\end{bmatrix}​100​21k​151​​ be invertible?

133 - FML 3 - 18.1W e.g. 10

 Find the value(s) of c such that A is NOT invertible given A=  [c−c1−1−1−11−cc]\bcb{\boldsymbol{ A =  \begin{bmatrix} c & -c & 1 \\ -1 & -1 & -1 \\ 1 & -c & c  \end{bmatrix} }}A=  ​c−11​−c−1−c​1−1c​​

Practice: Matrix Inverse

Find all values of k so that the matrix [−2003102−1k]\begin{bmatrix}
-2&0&0\\
3&1&0\\
2&-1&k
\end{bmatrix}​−232​01−1​00k​​ has an inverse.

Find the inverse of A=[101011110]A=\begin{bmatrix}
1&0&1\\
0&1&1\\1&1&0
\end{bmatrix}A=​101​011​110​​ .

For each of the following matrices, find the inverse or state why the inverse does not exist.
A=[1−1−2100012]A=\begin{bmatrix}
1&-1&-2\\
1&0&0\\
0&1&2
\end{bmatrix}A=​110​−101​−202​​
B=[3191506]B=\begin{bmatrix}
3&1&9\\
15&0&6
\end{bmatrix}B=[315​10​96​]

Practice: Matrix Inverse

Find all values of k for which the matrix [2−6−1k2−22]\begin{bmatrix}
2&-6\\
-1&k^2-22
\end{bmatrix}[2−1​−6k2−22​] has no inverse.

Matrix Inverse

Practice: Matrix Inverse
If [1ax2by0cz]\left[\begin{array}{c}
1&a&x\\
2&b&y\\
0&c&z
\end{array}\right]
​120​abc​xyz​​ is the inverse of the matrix[1p0−3q42r1]\left[\begin{array}{c}
1&p&0\\
-3&q&4\\
2&r&1
\end{array}\right]​1−32​pqr​041​​, find the value of rrr.

Let A=[214103421]A=\begin{bmatrix}2&1&4\\1&0&3\\4&2&1\end{bmatrix}A=​214​102​431​​. Find A−1A^{-1}A−1 if possible.

Find the inverse of [021255121]\begin{bmatrix}
0 &2& 1\\
2 &5 &5\\
1 &2 &1
\end{bmatrix}​021​252​151​​

The following matrix A depends on a parameter p. Find all the values of the parameter p for which the
 corresponding matrix is not invertible.
A=[123410002000300040010𝑝𝑝+1𝑝+20−𝑝−2𝑝+1−3𝑝+2]A=\begin{bmatrix}
1&2&3&4\\
1000&2000&3000&4001\\
0&𝑝&𝑝 + 1&𝑝 + 2\\
0&−𝑝&−2𝑝 + 1&−3𝑝 + 2

\end{bmatrix}A=​1100000​22000p−p​33000p+1−2p+1​44001p+2−3p+2​​

For what value or values of a (if any) is the matrix below invertible?
[1210350𝑎1]\begin{bmatrix}
1& 2& 1\\
0 &3& 5\\
0 &𝑎 &1
\end{bmatrix}​100​23a​151​​

The following matrix A depends on a parameter p. Find all the values of the parameter 
 p for which the corresponding matrix is not invertible.
A=[123410002000300040010𝑝𝑝+1𝑝+20−𝑝−2𝑝+1−3𝑝+2]A=\begin{bmatrix}
1&2&3&4\\
1000&2000&3000&4001\\
0&𝑝&𝑝 + 1&𝑝 + 2\\
0&−𝑝&−2𝑝 + 1&−3𝑝 + 2

\end{bmatrix}A=​1100000​22000p−p​33000p+1−2p+1​44001p+2−3p+2​​

Practice Question 3: Matrix Inverse

Practice Question: Matrix Inverse
For what value(s) of k will the matrix[1210150k1]\begin{bmatrix}
1&2&1\\
0&1&5\\
0&k&1
\end{bmatrix}​100​21k​151​​ be invertible?

Find all the values of k for which the matrix A has an inverse
A=[210102102k]A=\begin{bmatrix}
2&1&0\\
1&0&2\\
1&0&2k
\end{bmatrix}A=​211​100​022k​​

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]A=​201​11−1​302​​,

Find all the values of 𝑘 such that the matrix 𝐴=[33−610−k2]has an inverse.\text{Find all the values of 𝑘 such that the matrix 𝐴}=\begin{bmatrix}
3&3\\
-6&10-k^2
\end{bmatrix}\text{has an inverse.}Find all the values of k such that the matrix A=[3−6​310−k2​]has an inverse.

Matrix Inverse

If [1ax2by0cz]\left[\begin{array}{c}
1&a&x\\
2&b&y\\
0&c&z
\end{array}\right]
​120​abc​xyz​​ is the inverse of the matrix[1p0−3q42r1]\left[\begin{array}{c}
1&p&0\\
-3&q&4\\
2&r&1
\end{array}\right]​1−32​pqr​041​​, find the value of rrr.

Practice: Adjoint Matrix

Let B=[1−102111000−11−21−2−1]B=\left[\begin{array}{rrrr}
1&-1&0&2\\
1&1&1&0\\
0&0&-1&1\\
-2&1&-2&-1
\end{array}\right]B=​110−2​−1101​01−1−2​201−1​​, and note that det(B)=−1\text{det}(B)=-1det(B)=−1
Use the determinant and the adjoint of BBB to find the entry in position (4,3)(4,3)(4,3) of the matrix B−1B^{-1}B−1

Find the inverse of the matrix [10221−13−215]\begin{bmatrix}
1&0&2\\
2&1&-1\\
3&-2&15
\end{bmatrix}​123​01−2​2−115​​.

Matrix Inversion Algorithm

Example: Matrix Inverse Algorithm
Find the inverse of A=[1325]A=\left[\begin{array}{rr}1 & 3\\2 & 5\end{array}\right]A=[12​35​] using the formula for a 2×22\times22×2 matrix, and using the Gauss-Jordan elimination algorithm.

Find the inverse of the matrix A=[10221−13−215]A=\left[\begin{array}{rrr}
1&0&2\\
2&1&-1\\
3&-2&15
\end{array}\right]A=​123​01−2​2−115​​

Matrix operations

Suppose that  A=[1−103−20041]A=\begin{bmatrix}1 & -1 & 0\\3 & -2 & 0\\0 & 4 & 1\end{bmatrix}A=​130​−1−24​001​​and BBBis a 3×23 \times 23×2matrix. If the product AB=[34011−2]AB = \begin{bmatrix}3 & 4\\0 & 1\\1 & -2\end{bmatrix}AB=​301​41−2​​, find the (1,2)-entry of BBB(i.e. B12B_{12}B12​).

Invertible matrix

For what value(s) of aaawill the matrix [2−43a]\begin{bmatrix}2 & -4\\3 & a\end{bmatrix}[23​−4a​] be invertible?

Matrix Inverse

Practice: Matrix Inverse
If [1ax2by0cz]\left[\begin{array}{c}
1&a&x\\
2&b&y\\
0&c&z
\end{array}\right]
​120​abc​xyz​​ is the inverse of the matrix[1p0−3q42r1]\left[\begin{array}{c}
1&p&0\\
-3&q&4\\
2&r&1
\end{array}\right]​1−32​pqr​041​​, find the value of rrr.

Practice Question: Matrix Inverse

Practice Question: Matrix Inverse
If [1ax2by0cz]\left[\begin{array}{}
1&a&x\\
2&b&y\\
0&c&z
\end{array}\right]
​120​abc​xyz​​ is the inverse of the matrix[1p0−3q42r1]\left[\begin{array}{}
1&p&0\\
-3&q&4\\
2&r&1
\end{array}\right]​1−32​pqr​041​​, find the value of rrr.

Suppose that A=[1−1031−4]A=\begin{bmatrix}1 &-1&0\\3&1&-4\end{bmatrix}A=[13​−11​0−4​] and RRRis the reduced row echelon form of AAA. Find the invertible matrix UUUsuch that R=UAR=UAR=UA.

For what value(s) of aaawill the matrix [2−43a]\begin{bmatrix}2 & -4\\3 & a\end{bmatrix}[23​−4a​]be invertible?

Practice: Matrix Inverse

Find all values of k for which the matrix [2−6−1k2−22]\begin{bmatrix}
2&-6\\
-1&k^2-22
\end{bmatrix}[2−1​−6k2−22​] has no inverse.

Practice: Matrix Inverse

Practice Question: Matrix Inverse
Find all the values of kkk such that the matrix[210102102k]\begin{bmatrix}
2&1&0\\
1&0&2\\
1&0&2k
\end{bmatrix}​211​100​022k​​ has an inverse.

Practice: Matrix Inverse

Practice Question: Matrix Inverse
For what value(s) of k will the matrix[1210150k1]\begin{bmatrix}
1&2&1\\
0&1&5\\
0&k&1
\end{bmatrix}​100​21k​151​​ be invertible?

Practice: Matrix Inverse

Practice Question: Matrix Inverse
For what value(s) of kkk will the matrix[1210150k1]\left[\begin{array}{rrr}
1&2&1\\
0&1&5\\
0&k&1
\end{array}\right]​100​21k​151​​ be invertible?

Practice Question 1: Matrix Inverse

Practice Question: Matrix Inverse
Find all the values of kkk such that the matrix[210102102k]\begin{bmatrix}
2&1&0\\
1&0&2\\
1&0&2k
\end{bmatrix}​211​100​022k​​ has an inverse.

Practice: Matrix Inverse

Find all values of k so that the matrix [−2003102−1k]\begin{bmatrix}
-2&0&0\\
3&1&0\\
2&-1&k
\end{bmatrix}​−232​01−1​00k​​ has an inverse.

Practice: Matrix Inverse (~2016 Test2 #15)

Practice: Matrix Inverse
Find all the values of kkk such that the matrix[210102102k]\begin{bmatrix}
2&1&0\\
1&0&2\\
1&0&2k
\end{bmatrix}​211​100​022k​​ has an inverse.

Practice: Matrix Inverse

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