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Practice: Elementary Matrix

Related Topics

Wize University Linear Algebra Textbook > Matrices

Elementary Matrices

5 Activities

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Find an elementary matrix EE such that EB=CEB=C if:

B=[11300364]B=\left[\begin{array}{rr} 1&1\\ 3&0\\ 0&3\\ 6&4 \end{array}\right] and C=[101300364]C=\left[\begin{array}{rr} 10&1\\ 3&0\\ 0&3\\ 6&4 \end{array}\right]

More Elementary Matrices Questions:
Write down matrix A−1A^{-1} as the product of elementary matrices if:
A=[01211021−1]A=\begin{bmatrix}0&1&2\\1&1&0\\2&1&-1\end{bmatrix}
Write down matrix A−1A^{-1} as the product of elementary matrices if:
A=[01211021−1]A=\begin{bmatrix}0&1&2\\1&1&0\\2&1&-1\end{bmatrix}
Elementary matrices
Which of the following are elementary matrices?
A=[10001000−3]B=[0001001001001000]A = \begin{bmatrix} 1 & 0 & 0\\0 & 1 & 0\\0 & 0 & -3\end{bmatrix} \quad B = \begin{bmatrix} 0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\\1 & 0 & 0 & 0\end{bmatrix}
C=[001010100]D=[10−71]C = \begin{bmatrix} 0 & 0 & 1\\0 & 1 & 0\\1 & 0 & 0\end{bmatrix} \quad D = \begin{bmatrix} 1 & 0 \\-7 & 1 \end{bmatrix}
Example: Elementary Matrix
Find an elementary matrix EE such that A=EBA=EB if:
A=[246−1−1−11−13]A=\left[\begin{array}{rrr} 2&4&6\\ -1&-1&-1\\ 1&-1&3 \end{array}\right] and B=[246−1−1−1315]B=\left[\begin{array}{rrr} 2&4&6\\ -1&-1&-1\\ 3&1&5 \end{array}\right]
Which of the following are elementary matrices?
A=[10001000−3]B=[0001001001001000]A = \begin{bmatrix} 1 & 0 & 0\\0 & 1 & 0\\0 & 0 & -3\end{bmatrix} \quad B = \begin{bmatrix} 0 & 0 & 0 & 1\\0 & 0 & 1 & 0\\0 & 1 & 0 & 0\\1 & 0 & 0 & 0\end{bmatrix}
C=[001010100]D=[10−71]C = \begin{bmatrix} 0 & 0 & 1\\0 & 1 & 0\\1 & 0 & 0\end{bmatrix} \quad D = \begin{bmatrix} 1 & 0 \\-7 & 1 \end{bmatrix}
Practice Question: Matrix of an ERO
Let EE be the elementary matrix obtained from I2I_2 by multiplying R2 by 1/3. Let BB be any 2×22\times2 matrix. Show that EBEB is the result of multiplying R2 of BB by 1/3. Find the inverse of EE and show that it is an elementary matrix.
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} }
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}\!,