Problem 19 - College Algebra

Question

In Exercises \(19-24\), perform each matrix row operation and write the new matrix. \(\left[\begin{array}{rrr|r}2 & -6 & 4 & 10 \\ 1 & 5 & -5 & 0 \\ 3 & 0 & 4 & 7\end{array}\right] \quad \frac{1}{2} R_{1}\)

Step-by-Step Solution

Answer

The new matrix after the specified row operation is \(\left[\begin{array}{rrr|r}1 & -3 & 2 & 5 \ 1 & 5 & -5 & 0 \ 3 & 0 & 4 & 7\end{array}\right]\).

Step 1: Identify the row operation

The given operation \(\frac{1}{2} R_{1}\) means that every element in the first row of the matrix should be multiplied by \(\frac{1}{2}\).

Step 2: Perform the row operation

Multiplying every element of the first row by \(\frac{1}{2}\), i.e., \(\frac{1}{2}*2, \frac{1}{2}*(-6), \frac{1}{2}*4, \frac{1}{2}*10\), this gives new elements as \(1, -3, 2, 5\).

Step 3: Write the new matrix

Substitute the new elements into the first row of the matrix, the new matrix becomes \(\left[\begin{array}{rrr|r}1 & -3 & 2 & 5 \ 1 & 5 & -5 & 0 \ 3 & 0 & 4 & 7\end{array}\right]\).

Key Concepts

Understanding Matrices

Matrices are rectangular arrays of numbers arranged in rows and columns. They are often represented inside brackets, with each entry identified by its position within the array. For example, in a matrix with 3 rows and 3 columns, element a_ij is located at row i, column j. Matrices are widely used in various fields like computer graphics, statistics, and physics.

• Matrices can have one or many rows and columns.
• The size of a matrix is determined by its number of rows and columns (m x n).
• They can represent data, transformations, equations, and more.

In linear algebra, matrices are crucial tools because they simplify operations on complex equations. Each entry within a matrix can be manipulated through algebraic operations, helping solve systems of equations, among other tasks.

Exploring Elementary Row Operations

Elementary row operations are simple manipulations applied to the rows of a matrix. These operations are fundamental in solving systems of linear equations and transforming matrices into more convenient forms like row-echelon form. In the given exercise, we performed an elementary row operation by multiplying every element of the first row by a scalar (\(\frac{1}{2}\))

• Row Scaling: Multiply all elements in a row by a nonzero scalar.
• Row Replacement: Add a multiple of one row to another.
• Row Interchange: Swap two rows in the matrix.

These operations help in pivoting rows to achieve simpler matrix forms, allowing easier solving of linear equations. It's important to note these operations do not change the solution of the system they represent, just the expression of it.

Delving into Matrix Multiplication

Matrix multiplication involves a more complex process than simple element-wise multiplication. It requires multiplying the rows of the first matrix by the columns of the second, summing the products to form a new matrix. It is essential to ensure compatibility, where the number of columns in the first matrix should equal the number of rows in the second matrix.

• Each element in the resulting matrix is the dot product of a row from the first matrix and a column from the second matrix.
• The resulting matrix will have dimensions determined by the number of rows of the first matrix and the number of columns of the second matrix.
• Matrix multiplication is not commutative: \(AB eq BA\).

This operation is critical in various applications, notably in graphics transformations and system solving via linear algebraic methods.

The Role of Linear Algebra

Linear algebra is a branch of mathematics focusing on vectors, vector spaces, and linear transformations represented by matrices. It provides the framework for understanding and working with linear equations and transformations, vital in numerous scientific and engineering applications.

• Vector Spaces: Collections of vectors that can be added together and multiplied by scalars, adhering to certain rules.
• Linear Equations: Equations of the first degree that can be represented in matrix form, simplifying complex systems.
• Transformations: Operations that move or change vectors within a space, usually described using matrices.

Linear algebra's foundations allow us to manipulate equations and data sets to arrive at solutions efficiently. Knowing its principles, such as matrices and operations like those described, is essential for delving deeper into the field and applications like machine learning and quantum physics.