Question

A matrix \(A\) is given below. In Exercises 16 20, a matrix \(B\) is given. Give the row operation that transforms \(A\) into \(B\). $$A=\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 2 & 3\end{array}\right]$$ $$B=\left[\begin{array}{lll}3 & 5 & 7 \\ 1 & 0 & 1 \\ 1 & 2 & 3\end{array}\right]$$

Short answer

The row operation is \(R_1 = R_1 + 2R_3\).

Step-by-step solution

Understand the Matrix Formations

Matrix \(A\) and \(B\) are both \(3 \times 3\) matrices. To transform matrix \(A\) into matrix \(B\), observe that only the first row of matrix \(A\) is different from matrix \(B\). The second and third rows remain unchanged.

Determine the Required Row Operation

Matrix \(A\)'s first row is \([1, 1, 1]\). Matrix \(B\)'s first row is \([3, 5, 7]\). To match the elements, you'll need a combination of matrix \(A\)'s rows. Notice that: \(3 = 1 + 2 \times 1\), \(5 = 1 + 2 \times 2\), \(7 = 1 + 2 \times 3\). This suggests that to form the first row of \(B\), perform the operation: \(R_1 = R_1 + 2R_3\) on matrix \(A\).

Apply the Row Operation

Apply the row operation \(R_1 = R_1 + 2R_3\). Perform this operation on the first row of matrix \(A\): - The first element becomes \(1 + 2 \times 1 = 3\). - The second element becomes \(1 + 2 \times 2 = 5\). - The third element becomes \(1 + 2 \times 3 = 7\).Thus, the first row of matrix \(A\) is transformed into the first row of matrix \(B\).

Verify the Transformation

After performing the operation \(R_1 = R_1 + 2R_3\), the matrix becomes: \[\begin{bmatrix} 3 & 5 & 7 \ 1 & 0 & 1 \ 1 & 2 & 3 \end{bmatrix}\].Compare this with matrix \(B\). This transformation matches matrix \(B\), confirming the row operation is correct.

Key concepts

Matrix Operations

Matrix operations are fundamental techniques used to manipulate matrices to achieve desired results. These operations include addition, subtraction, multiplication, transposition, and finding determinants.
Understanding how to apply these operations to matrices is essential in various mathematical and engineering fields.

• Addition/Subtraction: Matrices of the same dimensions can be added or subtracted by adding or subtracting corresponding elements.
• Multiplication: Matrix multiplication involves the dot product of rows and columns, and it is crucial to note that the order matters in multiplication.
• Transposition: This operation involves flipping a matrix over its diagonal, effectively swapping rows and columns.
• Determinants: These provide scalar values that can offer insights into matrix properties such as singularity.

Mastering these operations allows for advanced applications such as solving linear systems and performing transformations.

Row Operations

Row operations are specific techniques used to manipulate the rows of a matrix. They are essential in solving linear equations and simplifying matrices to their row-echelon forms.
These operations include:

• Swapping rows: Interchange any two rows. Useful for rearranging matrices for easier calculations.
• Scaling rows: Multiply all elements of a row by a non-zero scalar, allowing adjustments of row magnitudes.
• Adding rows: Add or subtract a multiple of one row to another, enabling transformation into simpler forms.

In our original exercise, the operation applied was a row addition: specifically, multiplying a row by a scalar and adding it to another. This is crucial in transforming matrices for solutions and simplifying calculations.

3x3 Matrices

A 3x3 matrix comprises three rows and three columns, totaling nine elements. Due to their relatively small size, 3x3 matrices serve as excellent practice for learning matrix operations and serve numerous purposes in computing and mathematical problems.
The terms related to a 3x3 matrix include:

• Determinant: A scalar value representing certain properties of the matrix, such as invertibility. The determinant of a 3x3 can be calculated using a specific formula involving its elements.
• Inverse: Only possible when the determinant is non-zero. An inverse matrix can be used to solve systems of linear equations.
• Eigenvalues & Eigenvectors: Important in many fields such as physics and engineering, offering insights into the behavior of systems.

Understanding these concepts helps in solving more complex problems and applying 3x3 matrices to real-world scenarios.

Matrix Algebra

Matrix algebra involves the study and manipulation of matrices through various algebraic rules and operations. It forms the foundation of linear algebra and includes a range of techniques used to analyze matrix equations and systems.

• Systems of Linear Equations: Matrices can compactly represent and solve linear systems using methods like Gauss-Jordan elimination.
• Matrix Inverses: Similar to reciprocals in numbers, inverses help solve equations where matrices are multiplied.
• Linear Transformations: Matrices can represent linear transformations, which can modify vectors in space.

Studying matrix algebra enhances understanding of more complex mathematical procedures and allows the application of matrices beyond theoretical exercises, illustrating their utility in modeling and solving real-world problems.