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Inverse of a Matrix

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Wize University Linear Algebra Textbook > Matrices

Matrix Inverse

4 Activities

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Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
e.g. Prove that, for an invertible matrix and a constant cc, cA‾c\underline{ A} is also invertible, and  ⁣(cA‾)−1=c−1A‾−1 ⁣\!(c \underline{A})^{ −1} = c^{ −1}\underline{A}^{−1}\!.

More Matrix Inverse Questions:
Properties of Matrix Inverse
Suppose A,BA,B are any 2 n×nn\times n matrices, and that A+BA+B is invertible.
Show that A(A+B)−1B=B(A+B)−1AA(A+B)^{-1}B=B(A+B)^{-1}A .
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} }} then the (3,2)\bcb{\boldsymbol{ (3, 2)}} entry in adj(A)\bcb{\boldsymbol{ \text{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
Transpose of a Matrix
Prove that, for an invertible matrix A‾\bcb{\A} , A‾T\bcb{\mtran{\A}} is also invertible, and (AT)−1=(A−1)T\bcb{(A^T )^ {−1} = (A^{−1} )^ T}.
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
Practice Question: Matrix Properties
Practice Question: Matrix Properties
Let AA and BB be invertible matrices. Which of the following statements is/are always true?
i.) AB=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] is invertible, what is the third row of AA−1AA^{-1}?
Equation problem
Given that A2=(I−2A)(I+A)A^2=(I-2A)(I+A)and AAis invertible, which of the following is an expression for A−1A^{-1}?
Invertible Matrix
Let AAand BBbe square matrices and UUis an invertible matrix. Given that BU=ABU=A, which of the following statements is always true?
Given that A2=(I−2A)(I+A)A^2=(I-2A)(I+A)and AAis invertible, which of the following is an expression for A−1A^{-1}?
Let AAand BBbe square matrices and UUis an invertible matrix. Given that BU=ABU=A, which of the following statements is always true?
Let AAand BBbe n×nn\times nmatrices, and let kkbe 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} and BBis a 3×23\times2 matrix. If the product AB=[34011−2]AB = \begin{bmatrix}3 & 4\\0 & 1\\1 & -2\end{bmatrix}, find the (1,2)-entry of BB(i.e. B12B_{12}).
Practice: Matrix Inverse Application
Practice: Matrix Inverse Application
Let A=[120−1]A=\left[\begin{array}{c} 1&2\\0&-1 \end{array}\right] and suppose that 3AB−[1−120]=I23AB- \left[\begin{array}{c} 1&-1\\ 2&0 \end{array}\right] =I_2.
Find the matrix BB.
Practice: Matrix Properties (~2015 Test2 #16)
Practice: Matrix Properties
Let AA and BB be invertible matrices. Which of the following statements is/are always true? (select all that apply)
i.) AB=BAAB=BA
Transpose of a Matrix
e.g. Prove that, for an invertible matrix A,ATA, A^T is also invertible, and (AT)−1=(A−1)T(A^T )^ {−1} = (A^{−1} )^ T
Let AAand BBbe n×nn\times nmatrices, and let kkbe 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}, find the (3,3)-entry of A−1A^{-1}if 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] is the inverse of the matrix[1p0−3q42r1]\left[\begin{array}{c} 1&p&0\\ -3&q&4\\ 2&r&1 \end{array}\right], find the value of rr.
Practice: Matrix Inverse
Let AA, BB, CC and DD 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:
Matrix Multiplication
Express the following matrix A‾\bcb{ \A } as a product of five (or fewer) elementary matrices); repeat for A‾−1\bcb{ \minv{\A} }.
A‾=[−300201010]\bcb{ \A=\begin{bmatrix} -3&0&0\\2&0&1\\0&1&0 \end{bmatrix} }
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] is invertible, what is the third row of AA−1AA^{-1}?
Practice: Matrix Inverse
If A=[abcdefghi]A= \begin{bmatrix} a&b&c\\ d&e&f\\ g&h&i \end{bmatrix} is an invertible matrix and C=AA−1C=AA^{-1}, find the first row of CC .
Practice: Matrix Inverse
Let AA, BB, CC and DD be invertible matrices of size 10×1010\times10
Select the expression which is equivalent to the matrix:
(A(BC)−1D)T\left(A\left(BC\right)^{-1}D\right)^T
Practice: Matrix Inverse
Practice Question: Matrix Inverse
Let A=[1312]A=\begin{bmatrix} 1&3\\ 1&2 \end{bmatrix}. Find all 2×22\times2 matrices BB such that AB=[54−20].AB=\begin {bmatrix}5&4\\-2&0\end{bmatrix}.
Practice: Matrix Operations
Let AA, BB, CC and DD be invertible matrices of size 10×1010\times 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:
Matrix Multiplication
e.g. Express the following matrix A‾\underline{A} as a product of five (or fewer) elementary matrices); repeat for  ⁣A‾−1 ⁣\!\underline{A}^{−1}\!,
Invertible Matrix Theorem
Example: Invertible Matrix Theorem
Consider the linear system 𝐴𝑥⃗=𝑏⃗𝐴𝑥⃗=\vec{𝑏} where A5×5A_{5 \times 5} is an invertible matrix. What is rank([𝐴 ∣ b⃗]){\rm rank}([𝐴\ |\ \vec{b}])?