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Matrix Transpose

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Wize University Linear Algebra Textbook > Matrices

Matrix Transpose

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Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
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Prove that for any square matrix AA, the matrix A−ATA-A^T is skew-symmetric.
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}.
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}.
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}
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.