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Transpose of a Matrix

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Wize University Linear Algebra Textbook > Matrices

Matrix Transpose

5 Activities

Wize University Linear Algebra Textbook > Matrices

Matrix Inverse

4 Activities

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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}.


More Matrix Transpose Questions:
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 2 quiz ✓}$ | 133 - FML 1 - 18.1W e.g. 25
Show that A‾T  ⁣A‾ \bcb{\boldsymbol{ \mtran{\A}\ul{A}}} \, is a symmetric matrix for any n×n\bcb{\boldsymbol{ n \times n}} matrix   ⁣A‾\bcb{\boldsymbol{ \ul{A} }}.
Practice: Matrix Operations
Given that A=[−123−4]A=\begin{bmatrix} -1&2\\3&-4 \end{bmatrix} and B=[11−22]B=\begin{bmatrix}1&1\\-2&2\end{bmatrix}, let C=(3A−2BT+I2)TC=(3A-2B^T+I_2)^T. Determine C22C_{22}.
Practice Question 8: Matrix Transpose
Practice Question: Matrix Transpose
Given the following matrices, find the values of a,ba, b and cc such that −3A=BT-3A=B^T.
A=[13−60152]A=\begin{bmatrix} 1&3&-6\\0&15&2 \end{bmatrix} and B=[−3a−95b2c−6]B=\begin{bmatrix}-3&a\\-9&5b\\2c&-6\end{bmatrix}
Matrix Transpose
Prove that for any square matrix AA, the matrix A−ATA-A^T is skew-symmetric.
Example: Matrix Operations
Example:
Given the matrices 𝐴=[100−3−22]𝐴=\begin{bmatrix}1&0\\0&-3\\-2&2\end{bmatrix}, B=[30201−2]\text{B}=\begin{bmatrix}3&0\\2&0\\1&-2\end{bmatrix}, and C=[12010−1]C=\begin{bmatrix}1&2&0\\1&0&-1\end{bmatrix},
a) Find 2A−B2A-B.
Example: Matrix Transpose
Example:
Find the transpose of 𝐴=[1−3−2405]𝐴=\begin{bmatrix}1&-3\\-2&4\\0&5\end{bmatrix}, then state the sizes of AA and ATA^T.
Matrix Multiplication and Transpose
Suppose that A=[2−11103]A=\begin{bmatrix} 2&-1\\ 1&1\\ 0&3 \end{bmatrix} and B=[03−2−110]B=\begin{bmatrix} 0&3\\ -2&-1\\ 1&0 \end{bmatrix}.
Find 2ATB−(BTA)T−ATIB2A^TB-(B^TA)^T-A^TIB, where II is the 3x3 identity matrix.
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
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
Inverse of a Matrix
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}\!.
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}])?