Given the matrix 𝐴= 213 010 1-12 . a) find the inverse of 𝐴. b) solve the sy…

Solving Matrix Equations

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Wize University Linear Algebra Textbook > Matrices

Matrix Inverse

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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                                                                  

                                                                                       𝑦−3=−2𝑦−3=−2y−3=−2

                                                                                  𝑥−𝑦+2𝑧=4𝑥−𝑦+2𝑧=4x−y+2z=4

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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

Practice: Matrix Inverse

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​​

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​​

Properties of Matrix Inverse

Suppose A,BA,BA,B are any 2 n×nn\times nn×n matrices, and that A+BA+BA+B is invertible.
Show that A(A+B)−1B=B(A+B)−1AA(A+B)^{-1}B=B(A+B)^{-1}AA(A+B)−1B=B(A+B)−1A .

133 - FML 3 - 18.1W e.g. 75

   ⁣A‾=  [123−201−1−23]\bcb{\boldsymbol{ \ul{A} =  \begin{bmatrix} 1 & 2 & 3 \\ -2 & 0 & 1 \\ -1 & -2 & 3  \end{bmatrix} }}A​=  ​1−2−1​20−2​313​​ then the (3,2)\bcb{\boldsymbol{ (3, 2)}}(3,2) entry in adj(A)\bcb{\boldsymbol{ \text{adj}(A)}}adj(A) is:
(a) −4(b)  0(c)  4(d)  None of the above\text{(a)}\, \boldsymbol{  -4}
 \qquad\qquad\qquad
\text{(b)}\; \boldsymbol{  0}
 \qquad\qquad\qquad
\text{(c)} \;\boldsymbol{  4}
 \qquad\qquad\qquad
\text{(d)} \;\text{\bf{None of the above}}
 \qquad\qquad\qquad(a)−4(b)0(c)4(d)None of the above

Transpose of a Matrix

Prove that, for an invertible matrix A‾\bcb{\A}A​ , A‾T\bcb{\mtran{\A}}A​T is also invertible, and (AT)−1=(A−1)T\bcb{(A^T )^ {−1} = (A^{−1} )^ T}(AT)−1=(A−1)T.

Practice Question: Matrix Properties

Practice Question: Matrix Properties
Let AAA and BBB be invertible matrices. Which of the following statements is/are always true?
i.) AB=BAAB=BAAB=BA

Practice Question: Matrix Inverse

Practice Question: Matrix Inverse
If A=[abcxyz10−1]A=\left[\begin{array}{l}
a&b&c\\
x&y&z\\
1&0&-1
\end{array}\right]A=​ax1​by0​cz−1​​ is invertible, what is the third row of AA−1AA^{-1}AA−1?

Equation problem

Given that A2=(I−2A)(I+A)A^2=(I-2A)(I+A)A2=(I−2A)(I+A)and AAAis invertible, which of the following is an expression for A−1A^{-1}A−1?

Invertible Matrix

Let AAAand BBBbe square matrices and UUUis an invertible matrix. Given that BU=ABU=ABU=A, which of the following statements is always true?

Given that A2=(I−2A)(I+A)A^2=(I-2A)(I+A)A2=(I−2A)(I+A)and AAAis invertible, which of the following is an expression for A−1A^{-1}A−1?

Let AAAand BBBbe square matrices and UUUis an invertible matrix. Given that BU=ABU=ABU=A, which of the following statements is always true?

Let AAAand BBBbe n×nn\times nn×nmatrices, and let kkkbe a constant. Which of the following statements is always true?

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\times23×2 matrix. 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​).

Practice: Matrix Inverse Application

Practice: Matrix Inverse Application
Let A=[120−1]A=\left[\begin{array}{c}
1&2\\0&-1
\end{array}\right]A=[10​2−1​] and suppose that 3AB−[1−120]=I23AB-

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

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

Practice: Matrix Properties (~2015 Test2 #16)

Practice: Matrix Properties
Let AAA and BBB be invertible matrices. Which of the following statements is/are always true? (select all that apply)
i.) AB=BAAB=BAAB=BA

Inverse of a Matrix

e.g. Prove that, for an invertible matrix  and a constant ccc,  cA‾c\underline{ A}cA​ is also invertible, and  ⁣(cA‾)−1=c−1A‾−1 ⁣\!(c \underline{A})^{ −1} = c^{ −1}\underline{A}^{−1}\!(cA​)−1=c−1A​−1.

Transpose of a Matrix

e.g. Prove that, for an invertible matrix A,ATA, A^TA,AT is also invertible, and (AT)−1=(A−1)T(A^T )^ {−1} = (A^{−1} )^ T(AT)−1=(A−1)T 

Let AAAand BBBbe n×nn\times nn×nmatrices, and let kkkbe a constant. Select the statement that is always true.

Given thatA=[3−32−2461−10]A=\begin{bmatrix}3 & -3 & 2\\-2 & 4 & 6\\1 & -1 & 0\end{bmatrix}A=​3−21​−34−1​260​​, find the (3,3)-entry of A−1A^{-1}A−1if it exists. 

Practice: Matrix Inverse (~2016 Test2 #17)

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: Matrix Inverse

Let AAA, BBB, CCC and DDD be size 10 by 10 invertible matrices: Select the expression which is equal to the matrix (A(BC)−1D)T\left(A\left(BC\right)^{-1}D\right)^T(A(BC)−1D)T:

Matrix Multiplication

Express the following matrix A‾\bcb{ \A } A​ as a product of five (or fewer) elementary matrices); repeat for A‾−1\bcb{ \minv{\A} }A​−1.
A‾=[−300201010]\bcb{
\A=\begin{bmatrix}
-3&0&0\\2&0&1\\0&1&0
\end{bmatrix}
}A​=​−320​001​010​​

Practice: Matrix Inverse (~2015 Test2 #17)

Practice: Matrix Inverse
If A=[abcxyz10−1]A=\left[\begin{array}{l}
a&b&c\\
x&y&z\\
1&0&-1
\end{array}\right]A=​ax1​by0​cz−1​​ is invertible, what is the third row of AA−1AA^{-1}AA−1?

Practice: Matrix Inverse

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 .

Practice: Matrix Inverse

Let AAA, BBB, CCC and DDD be invertible matrices of size 10×1010\times1010×10
Select the expression which is equivalent to the matrix:
(A(BC)−1D)T\left(A\left(BC\right)^{-1}D\right)^T(A(BC)−1D)T

Practice: Matrix Inverse

Practice Question: Matrix Inverse
Let A=[1312]A=\begin{bmatrix}
1&3\\
1&2
\end{bmatrix}A=[11​32​]. Find all 2×22\times22×2 matrices BBB such that AB=[54−20].AB=\begin
{bmatrix}5&4\\-2&0\end{bmatrix}.AB=[5−2​40​].

Practice: Matrix Operations

Let AAA, BBB, CCC and DDD be invertible matrices of size 10×1010\times 1010×10
Show that: (A(BC)−1D)T=DT(BT)−1(CT)−1AT\left(A\left(BC\right)^{-1}D\right)^T=D^T\left(B^{T}\right)^{-1}\left(C^{T}\right)^{-1}A^T(A(BC)−1D)T=DT(BT)−1(CT)−1AT:

Matrix Multiplication

e.g. Express the following matrix A‾\underline{A}A​ as a product of five (or fewer) elementary matrices); repeat for  ⁣A‾−1 ⁣\!\underline{A}^{−1}\!A​−1,

Invertible Matrix Theorem

Example: Invertible Matrix Theorem
Consider the linear system 𝐴𝑥⃗=𝑏⃗𝐴𝑥⃗=\vec{𝑏}Ax⃗=b where A5×5A_{5 \times 5}A5×5​ is an invertible matrix. What is rank([𝐴 ∣ b⃗]){\rm rank}([𝐴\ |\ \vec{b}])rank([A ∣ b])?

Given the matrix 𝐴= 213 010 1-12 . a) find the inverse of 𝐴. b) solve the sy…

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