Question

Perform the given row operations on \(A,\) where $$A=\left[\begin{array}{ccc}2 & -1 & 7 \\ 0 & 4 & -2 \\ 5 & 0 & 3\end{array}\right]$$ $$-1 R_{1} \rightarrow R_{1}$$

Short answer

The updated matrix is: \( \begin{bmatrix} -2 & 1 & -7 \\ 0 & 4 & -2 \\ 5 & 0 & 3 \end{bmatrix} \).

Step-by-step solution

Understand the Row Operation

The given operation is \(-1 R_{1} \rightarrow R_{1}\).This means we need to multiply every element of the first row of matrix \(A\) by \(-1\). This operation will only affect the first row of the matrix.

Apply the Row Operation

To perform the operation \(-1 R_{1} \rightarrow R_{1}\), multiply each element in the first row by \(-1\):- For the element in the first column: \(2 \times (-1) = -2\)- For the element in the second column: \(-1 \times (-1) = 1\)- For the element in the third column: \(7 \times (-1) = -7\)So, the new first row will be \([-2, 1, -7]\).

Write the Updated Matrix

Insert the new first row into the matrix. The second and third rows remain unchanged. The new matrix after the row operation is:\[\begin{bmatrix}-2 & 1 & -7 \0 & 4 & -2 \5 & 0 & 3\end{bmatrix}\].

Key concepts

Matrix manipulation

Matrix manipulation involves changing the elements of a matrix to alter its structure in a way that reveals more information or makes it easier to work with. In the context of row operations, we manipulate matrices by changing rows as instructed. Row operations, such as scaling rows, swapping rows, and adding multiples of one row to another, help in solving systems of linear equations, finding inverses, and simplifying matrices. Matrix manipulation is fundamental in linear algebra and computational applications:

• **Simplification**: Matrices can be transformed for easier computation, such as reducing a matrix to row-echelon form.
• **Representation**: In computer graphics, matrices are used to represent transformations like rotations and scaling.
• **Optimization**: Manipulating matrices can reduce complexity, which is crucial in optimization problems.

Overall, matrix manipulation allows us to utilize matrices effectively across various fields, fostering problem-solving and creative solutions.

Elementary row operations

Elementary row operations are the tools of matrix manipulation that allow us to systematically change rows to achieve a desired form. There are three primary types of elementary row operations:

• **Row swapping**: Interchanging two rows within a matrix.
• **Row multiplication**: Multiplying all elements of a row by a non-zero constant, such as in \(-1 R_1 \rightarrow R_1\), which changes the sign of every element.
• **Row addition**: Adding a multiple of one row to another row, useful in eliminating variables.

When applying these operations, the key is maintaining the matrix's equivalency with respect to the linear system it represents. These operations are reversible and preserve the solutions of the corresponding systems of linear equations. As a result, elementary row operations are crucial in transforming matrices into more manageable forms, like row-echelon form or reduced row-echelon form, frequently used in solving linear equation systems and finding inverses.

Matrix algebra

Matrix algebra involves the study and manipulation of matrices through various operations like addition, subtraction, and multiplication. Understanding these algebraic rules and operations is essential for solving problems involving matrices and for applications in different scientific fields. Some basic concepts in matrix algebra include:

• **Matrix addition and subtraction**: Matrices are added or subtracted by performing element-wise addition or subtraction, which requires that the matrices have the same dimensions.
• **Scalar multiplication**: Every element of a matrix is multiplied by a scalar, modifying the matrix's magnitude.
• **Matrix multiplication**: A more complex operation that involves multiplying the rows of the first matrix by the columns of the second. The number of columns in the first matrix must match the number of rows in the second.
• **Identity matrix and inverse**: The identity matrix acts as the multiplicative identity in matrix algebra. An inverse matrix is a matrix that, when multiplied by the original, results in the identity matrix.

Matrix algebra serves as a foundation for numerous advanced concepts in mathematics and applications in physics, engineering, and computer science.