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Chapter 6: Problem 53

In your own words, describe each of the three matrix row operations. Give an example with each of the operations.

Short Answer

Expert verified
There are three types of matrix row operations. First, swapping any two rows in a matrix. Second, multiplying a row by a non-zero scalar. Lastly, adding or subtracting a multiple of one row to another row.

Step by step solution

01

Description and Example of Operation One

The first matrix row operation is swapping any two rows in a matrix. If we have matrix A as : \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \], we can swap the two rows to have: \[ \begin{bmatrix} c & d \ a & b \end{bmatrix} \]
02

Description and Example of Operation Two

The second type of matrix row operation is multiplying a row by a non-zero scalar. In case we have a matrix A as : \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \], we can multiply the second row by 2 to get: \[ \begin{bmatrix} a & b \ 2c & 2d \end{bmatrix} \]
03

Description and Example of Operation Three

The third type of matrix row operation is adding or subtracting a multiple of one row to another row. For instance, if we have a matrix A as : \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \], we can subtract the first row from the second to get: \[ \begin{bmatrix} a & b \ c-a & d-b \end{bmatrix} \]

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Most popular questions from this chapter

Explain how to evaluate a third-order determinant.

Consider the system $$\left\\{\begin{array}{l}a_{1} x+b_{1} y-c_{1} \\\a_{2} x+b_{2} y-c_{2}\end{array}\right.$$ Use Cramer's Rule to prove that if the first equation of the system is replaced by the sum of the two equations, the resulting system has the same solution as the original system.

Let $$\begin{aligned}&A=\left[\begin{array}{ll}1 & 0 \\\0 & 1\end{array}\right], \quad B=\left[\begin{array}{rr}1 & 0 \\\0 & -1\end{array}\right], \quad C=\left[\begin{array}{rr}-1 & 0 \\ 0 & 1\end{array}\right]\\\&D=\left[\begin{array}{rr}-1 & 0 \\\0 & -1\end{array}\right]\end{aligned}$$ Use any three of the matrices to verify a distributive property.

Find (if possible) the following matrices: \(a, A B\) \(\boldsymbol{b}, B A\) $$A=\left[\begin{array}{ll}4 & 2 \\\6 & 1 \\\3 & 5\end{array}\right], \quad B=\left[\begin{array}{rrr}2 & 3 & 4 \\\\-1 & -2 & 0\end{array}\right]$$

Find \(A^{-1}\) and check. $$A=\left[\begin{array}{rr}e^{x} & e^{3 x} \\\\-e^{3 x} & e^{5 x}\end{array}\right]$$
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