Problem 29 - Precalculus: Functions and Graphs

Question

If \(A=\left(a_{i j}\right)\) and \(B=\left(b_{i j}\right)\) are arbitrary square matrices of order \(2,\) show that \(|A B|=|A \| B|\)

Step-by-Step Solution

Answer

For 2x2 matrices, \(|AB| = |A||B|\) by determinant properties.

Step 1: Understand the Problem

We need to prove that the determinant of the product of two 2x2 matrices \( A \) and \( B \) is equal to the product of their determinants, \(|A B| = |A| |B|\).

Step 2: Recall the Determinant Property

Recall that for any square matrices \( A \) and \( B \), the determinant of their product \( AB \) is equal to the product of their determinants: \(|AB| = |A| |B|\). This is a fundamental property of determinants and holds for matrices of all orders.

Step 3: Calculate Determinant of a 2x2 Matrix

For a 2x2 matrix \( A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \), the determinant \(|A|\) is calculated as \(a_{11}a_{22} - a_{12}a_{21}\). Similarly, for \( B = \begin{pmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{pmatrix} \), \(|B| = b_{11}b_{22} - b_{12}b_{21}\).

Step 4: Derive Determinant of AB

If you multiply matrices \( A \) and \( B \) to find \( AB \), the resulting matrix is: \[AB = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{pmatrix} = \begin{pmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{pmatrix}\]The determinant of this matrix \( |AB| \) is calculated as:\[(a_{11}b_{11} + a_{12}b_{21})(a_{21}b_{12} + a_{22}b_{22}) - (a_{11}b_{12} + a_{12}b_{22})(a_{21}b_{11} + a_{22}b_{21})\]

Step 5: Simplify the Expression

Simplify the expression from the previous step to show that it equals \(|A| |B|\). This step involves straightforward algebraic simplification and confirmations that lead to the formula stating \(|A B| = (a_{11}a_{22} - a_{12}a_{21})(b_{11}b_{22} - b_{12}b_{21}) = |A||B|\).

Key Concepts

Determinant Properties

The determinant of a matrix reveals much more than just a number. It tells us important characteristics about the matrix itself. One of the crucial properties is that the determinant of a product of two square matrices is the same as the product of their determinants. So, if you have matrices \(A\) and \(B\), then the property is: \(|AB| = |A| |B|\).
This determinant property is fundamental and holds for matrices of any order, not just for 2x2 matrices. It's useful in various areas like solving equations and understanding transformations. Knowing how determinants work allows you to break down complex matrix operations into simpler ones that are easier to handle.

2x2 Matrices

2x2 matrices are some of the simplest matrices you will encounter. They are easy to work with and perfect for illustrating basic matrix properties. A 2x2 matrix looks like this: \(A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \).
To compute the determinant of a 2x2 matrix, you apply a simple formula: \(|A| = a_{11}a_{22} - a_{12}a_{21} \). This formula takes elements of the matrix and combines them to give you a single, insightful number. The computed determinant can tell you if the matrix is invertible, in which case the determinant will not be zero.

Matrix Multiplication

Matrix multiplication involves a process different from regular arithmetic multiplication. When multiplying two matrices, like \(A\) and \(B\), their size must be compatible; the number of columns in \(A\) must match the number of rows in \(B\). For 2x2 matrices, multiplication results in another 2x2 matrix.
The product \(AB\) is calculated by taking the rows of the first matrix \(A\) and multiplying them by the columns of the second matrix \(B\). So for two matrices \(A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix}\) and \(B = \begin{pmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{pmatrix}\), the multiplication results in a matrix:
\(AB = \begin{pmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{pmatrix} \).
This matrix product is significant in many mathematical and practical applications, from linear transformations to solving systems of equations.

Algebraic Simplification

Algebraic simplification plays a vital role in resolving seemingly complex expressions into more manageable forms. In the context of finding determinants of matrix products, simplification can confirm known properties.
For instance, the determinant of a product \( |AB| \) requires expanding and combining terms, where each part needs careful handling. The result from simplifying the expression \( (a_{11}b_{11} + a_{12}b_{21})(a_{21}b_{12} + a_{22}b_{22}) - (a_{11}b_{12} + a_{12}b_{22})(a_{21}b_{11} + a_{22}b_{21}) \) should eventually simplify to \((a_{11}a_{22} - a_{12}a_{21})(b_{11}b_{22} - b_{12}b_{21})\).
This simplification shows that \(|AB| = |A| |B|\), reinforcing the properties of determinants. Mastering algebraic simplification helps in verifying and understanding the deeper connections between mathematics and its underlying principles.