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Linear Algebra: Row and Column Operations

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Linear Algebra: Row and Column Operations

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Questions and Answers

What is the result of performing the operation $R_1 \to 4R_1$ on the matrix $A = \begin{bmatrix} 2 & 0 \ 3 & 4 \end{bmatrix}$?

\begin{bmatrix} 2 & 0 \\ 12 & 4 \end{bmatrix}

\begin{bmatrix} 4 & 0 \\ 3 & 4 \end{bmatrix}

\begin{bmatrix} 0 & 2 \\ 3 & 4 \end{bmatrix}

\begin{bmatrix} 8 & 0 \\ 3 & 4 \end{bmatrix} (correct)

If the operation $C_2 \to -3C_2$ is performed on $A = \begin{bmatrix} 2 & 0 \ 3 & 4 \end{bmatrix}$, what is the resulting matrix?

\begin{bmatrix} 2 & 0 \\ -9 & 4 \end{bmatrix}

\begin{bmatrix} 2 & -12 \\ 3 & 4 \end{bmatrix}

\begin{bmatrix} 2 & 0 \\ 3 & -12 \end{bmatrix} (correct)

\begin{bmatrix} 0 & -12 \\ 3 & 4 \end{bmatrix}

What does the operation $R_i \to R_i + kR_j$ accomplish?

It multiplies row $R_i$ by a constant.

It swaps row $R_i$ with row $R_j$.

It replaces row $R_i$ with the sum of rows $R_i$ and $R_j$.

It adds multiples of row $R_j$ to row $R_i$. (correct)

Which operation modifies a specific column of the matrix?

<p>$C_j \to kC_j$</p> Signup and view all the answers

What will be the result of the operation $C_i \to C_i + kC_j$?

<p>Elements of column $C_j$ are added to the corresponding elements of column $C_i$.</p> Signup and view all the answers

Study Notes

Row Operations

Multiplying a row by a constant:
- $R_i \to kR_i$
- Multiply each element in row $R_i$ by the constant $k$.
Adding a multiple of a row to another row:
- $R_i \to R_i + kR_j$
- Multiply each element in row $R_j$ by the constant $k$ and add the result to the corresponding element in row $R_i$.

Column Operations

Multiplying a column by a constant:
- $C_j \to kC_j$
- Multiply each element in column $C_j$ by constant $k$.
Adding a multiple of a column to another column:
- $C_i \to C_i + kC_j$
- Multiply each element in column $C_j$ by the constant $k$ and add the result to the corresponding element in column $C_i$.

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Description

This quiz focuses on row and column operations in linear algebra, including multiplying rows and columns by constants and adding multiples of rows or columns to others. Understanding these operations is essential for solving matrix equations and performing transformations. Test your knowledge through various questions related to these concepts.