Problem 79 - Algebra and Trigonometry Real Mathematics, Real People

Question

A property of determinants is given \((A \text { and } B\) are square matrices). State how the property has been applied to the given determinants and use a graphing utility to verify the results. If \(B\) is obtained from \(A\) by multiplying a row of \(A\) by a nonzero constant \(c\) or by multiplying a column of \(A\) by a nonzero constant \(c,\) then \(|B|=c|A|\) $$\begin{array}{l|ll} \text { (a) } & \begin{array}{ll} 1 & 5 \\ 6 & 9 \end{array}|=3| \begin{array}{ll} 1 & 5 \\ 2 & 3 \end{array} | \end{array}$$ $$\text { (b) }\left|\begin{array}{ll} 2 & 8 \\ 6 & 8 \end{array}\right|=8\left|\begin{array}{ll} 1 & 2 \\ 3 & 2 \end{array}\right|$$

Step-by-Step Solution

Answer

The property of determinants holds true for both the cases as evidenced by comparing the determinants of both sides of the given equations. The graphing utility also confirms the results. For both parts (a) and (b), the left-side determinant equals the right-side determinant.

Step 1: Understanding the property

Understand that when taking a matrix A and multiplying any of its rows or columns by a non-zero constant c, the determinant of the resulting matrix B is c times the determinant of the original matrix A, mathematically expressed as |B| = c|A|.

Step 2: Apply the determinant rule

The properties of determinants can be applied to verify the equation for both (a) and (b). The operation that transforms A into B involves multiplying a row by a non-zero constant. Take the matrices given and check if the property holds by calculating the determinant for both sides of the equation.

Step 3: Calculate the determinant for part (a)

For part (a), calculate determinant of right side matrix A as \( (1*3)-(5*2) = -7 \), and multiply with 3 to get final determinant as -21. Similarly, for B matrix, compute determinant as \( (1*9)-(5*6) = -21 \). Comparing, both are equal.

Step 4: Calculate the determinant for part (b)

For part (b), calculate determinant of right side matrix A as \( (1*2)-(2*3) = -4 \), and multiply with 8 to get final determinant as -32. Similarly, for B matrix, compute determinant as \( (2*8)-(8*6) = -32 \). Comparing, both are equal.

Step 5: Verify with a graphing utility

Now that the calculations have been completed, use a graphing utility to plot the equations. The equality of these results with those from the manual calculations verifies the property.

Key Concepts

Matrices

Understanding matrices is foundational in linear algebra. A matrix is essentially a rectangular array of numbers arranged in rows and columns. It's like a grid you might see in a spreadsheet, and every number (or "element") in the matrix is identified with a position based on its row and column location.

Some things to remember about matrices:

• A matrix can be of any size, denoted as "m x n", where "m" is the number of rows and "n" is the number of columns.
• When a matrix has the same number of rows and columns, it is called a "square matrix".
• Square matrices are particularly important when it comes to calculating determinants, which only exist for these matrices.

Square matrices have unique properties, especially in operations like determinant calculation and matrix multiplication, which are crucial for solving equations and understanding geometry and physics.

Determinant Properties

Determinants are special numbers that can be calculated from square matrices. The determinant gives important information about a matrix, such as whether it is invertible, meaning that you can "reverse" it back to the identity matrix.

Some key properties of determinants include:

• If any row or column of a matrix is multiplied by a constant, the determinant of the matrix is also multiplied by that constant.
• If two rows or columns in a matrix are the same or proportional, the determinant of the matrix is zero.
• If you swap two rows (or columns) in a matrix, the sign of the determinant changes.

In the exercise you studied, these properties help in understanding how the multiplication of a row or a column by a constant affects the determinant—essentially amplifying it by that same constant without altering its fundamental traits.

Matrix Multiplication

Matrix multiplication is a way of combining two matrices to produce a third matrix. This operation is not as straightforward as multiplying numbers, but it is an essential part of linear algebra.

Here's some vital information about matrix multiplication:

• The number of columns in the first matrix must match the number of rows in second one for the multiplication to be possible.
• The resulting matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.
• To multiply matrices, consider each element of the resulting matrix as the dot product of a row from the first matrix and a column from the second matrix.

In applications, the concept of matrix multiplication helps in areas like transformations and changes of coordinates in space. It's crucial to note that matrix multiplication is not commutative, meaning that \( AB eq BA \) in most cases. Understanding how matrix multiplication works is essential for grasping more complex mathematical concepts that build upon it.