Problem 64 - Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

(a) \(|A| = 0\), (b) \(|B| = 1\), (c) \(AB = \left[ \begin{array}{r} -2 & -4 \ 4 & 2 \end{array} \right]\), (d) \(|AB| = 4\)

Step 1: Calculate Determinant of Matrix A

Determinant of matrix A can be calculated using the formula \(ad - bc\). Therefore, \(|A| = -2*-2 - 4*1 = 4 - 4 = 0\).

Step 2: Calculate Determinant of Matrix B

Similarly, determinant of matrix B can be calculated using the formula \(ad - bc\). Therefore, \(|B| = 1*-1 - 0*2 = 1\).

Step 3: Calculate Matrix AB

Matrix multiplication AB will be done row by row from matrix A with column by column from matrix B. So, the first element of matrix AB is obtained by multiplying the first row of A with the first column of B, i.e., \(-2*1 + 4*0 = -2\). Repeat this process for other elements. The final matrix AB becomes \[ \left[ \begin{array}{cc} -2 & -4 \ 4 & 2 \end{array} \right] \].

Step 4: Calculate Determinant of Matrix AB

Again, determinant of matrix AB can be calculated using the formula \(ad - bc\). Therefore, \(|AB| = -2*2 - -4*4 = 4\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental concept in mathematics. It involves combining two matrices to produce a third matrix. When multiplying a matrix, the number of columns in the first matrix must equal the number of rows in the second matrix for the multiplication to be valid.

Here are some key points to remember about matrix multiplication:

• It's not commutative, meaning that if you swap the matrices, you may get a different result, i.e., generally, \(AB eq BA\).
• The resulting matrix will have dimensions equal to the number of rows from the first matrix and the number of columns from the second matrix.
• You multiply the elements of the rows of the first matrix by the corresponding elements in the columns of the second matrix and then sum those products to get each element of the resulting matrix.

To perform matrix multiplication with matrices \(A\) and \(B\) provided in the exercise, begin with the first row of matrix \(A\) against the first column of \(B\). Continue this row-by-column multiplication to fill each element of resulting matrix \(AB\).
Understanding these steps is crucial for carrying out successful matrix multiplication.

Matrix Arithmetic

Matrix arithmetic covers the operations of addition, subtraction, and multiplication applied to matrices. It extends concepts of basic arithmetic to more complex, multi-valued systems.

The perplexity of matrix arithmetic lies not in counting individual numbers but manipulating entire arrays. Here are a few highlights:

• Addition & Subtraction - Matrices can be added or subtracted if they have the same dimensions. You simply add or subtract their corresponding elements.
• Multiplication - As explained in the previous section, involves an array-by-array operation, producing a new matrix.
• Scalar Multiplication - Each element of the matrix is multiplied by the number (scalar) outside the matrix to produce a scaled matrix.

Matrix arithmetic is foundational for solving systems of equations, transformation problems in geometry, and operations in other scientific calculations.
In our exercise, we saw matrix multiplication, a prime example of matrix arithmetic. However, the determinant was not directly involved in arithmetic but in determining properties like matrix inversibility.

2x2 Matrix

A 2x2 matrix is a simple, square matrix with two rows and two columns. It's represented as:
\[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]
Each element \(a, b, c,\) and \(d\) is a scalar, which could be any real or complex number.

Key aspects of a 2x2 matrix include:

• Determinant - Calculated using the formula \(ad - bc\), it's a vital value that can determine if a matrix can be inverted. A determinant of zero indicates the matrix is not invertible.
• Simplicity - Being the smallest square matrix, it is often used to introduce matrix concepts, such as multiplication and determinants, before moving to larger matrices.
• Applications - Used in various computations in physics, engineering, and computer graphics for transformations and other mathematical operations.

In the given exercise, both matrices \(A\) and \(B\) are 2x2 matrices. This simplicity facilitates understanding how determinants are computed and how matrix multiplication transforms them into a new matrix.
Grasping the properties of a 2x2 matrix is an essential step before exploring more complex structures in linear algebra.