Problem 70 - Precalculus with Limits

Question

In Exercises 63-70, find (a) \(|A|\), (b) \(|B|\), (c) \(AB\), and (d) \(|AB|\). \(A = \left[ \begin{array}{r} 2 & 0 && 1 \\ 1 & -1 && 2 \\ 3 & 1 && 0 \end{array} \right]\), \(B = \left[ \begin{array}{r} 2 & -1 && 4 \\ 0 & 1 && 3 \\\ 3 & -2 && 1 \end{array} \right]\)

Step-by-Step Solution

Answer

|A| = 6, |B| = -6, AB = \[ \begin{array}{r} 7 & -4 && 11 \ 4 & -1 && 2 \ 7 & -3 && 15 \end{array} \], |AB| = 16

Step 1: Calculating Magnitude of Matrix A

The magnitude or determinant of a matrix, in this case A, can be calculated by the formula \(-a(ei−fh)+b(di−fg)+c(dh−eg)\). Here a, b, c represent the first row of matrix A; d, e, f represent the second row and g, h, i represent the third row. Plugging in values from matrix A, we get \(−2(2*0−1*1)+0(1*0−3*2)+1(1*1−3*−1)=6\).

Step 2: Calculating Magnitude of Matrix B

Using the same method as step 1, we calculate the magnitude of matrix B. After inserting the correct values, the determinant value is found to be \(-2(3*-2 - -1*1)+-1(2*3 - 0*1)+4(2*-2 - 3*0) = -6.\)

Step 3: Calculating the product AB

The next step is to calculate the product of matrices A and B. This is done by multiplying each row in the first matrix by each column in the second matrix and summing the results to form a new matrix. After computing, the matrix AB will be \[ \begin{array}{r} 7 & -4 && 11 \ 4 & -1 && 2 \ 7 & -3 && 15 \end{array} \]

Step 4: Calculating Magnitude of Matrix AB

By applying determinant formula as we did in step 1 and 2, this time on the product of A and B, we can determine |AB|. The determinant of AB is: \(-7(2*15 - -1*3)+-4(7*3 - 4*15)+11(7*-1 - 4*3)=16.\)

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental concept in linear algebra where you multiply two matrices to get a third matrix. The process involves a systematic method to calculate each element of the resulting matrix. The multiplication isn't as straightforward as multiplying numbers, as it requires a careful alignment of rows and columns.

• To multiply two matrices, ensure the number of columns in the first matrix matches the number of rows in the second matrix.
• Multiply each element of the rows of the first matrix with the corresponding elements of the columns of the second matrix.
• Sum up these products to form an element in the resulting matrix.

For instance, multiplying matrix A by matrix B involves taking the dot product of rows of A with columns of B. A key point to remember is that matrix multiplication is not commutative, meaning \( AB eq BA \) in general. This characteristic underscores the importance of maintaining specific order in multiplication.

Matrix Algebra

Matrix algebra is essentially the set of rules and operations that you can perform on matrices. These operations form the foundation for solving systems of linear equations, among other applications. Here, operations such as addition, subtraction, and multiplication of matrices follow specific rules.

• Matrix addition and subtraction requires that matrices have the same dimensions. This means each element in the matrices can be directly added or subtracted.
• Matrix multiplication, as discussed, involves a row-by-column approach, requiring compatible matrices.
• Scalar multiplication involves multiplying each element of a matrix by a scalar (single number).

While matrices find use in many computations, matrix algebra on its own is a robust tool for structuring, computing, and simplifying complex mathematical models. Understanding these operations grants the ability to manipulate matrices effectively and solve real-world problems.

Determinant Calculation

The determinant is a special number that can be calculated from a square matrix. It provides critical information about the matrix, including whether it is invertible. A non-zero determinant confirms that the matrix is invertible, while a zero determinant indicates it is not.

You calculate the determinant of a 3x3 matrix using a specific formula: \[ -a(ei−fh)+b(di−fg)+c(dh−eg)\]where \(a, b, c\) are the elements of the first row, \(d, e, f\) of the second row, and \(g, h, i\) of the third row.

• The determinant for a 2x2 matrix is simpler, calculated by \(ad - bc\).
• The value provides insights into the solutions of linear systems represented by the matrix.

Determinants also play a role in understanding the geometric transformations represented by matrices—like scaling or rotating—and are foundational in fields such as physics, engineering, and computer science.