Problem 21 - Precalculus: Functions and Graphs

Question

Verify the Identity by expanding each determinant. $$\left|\begin{array}{ll} a & b \\ c & d \end{array}\right|=-\left|\begin{array}{ll} c & d \\ a & b \end{array}\right|$$

Step-by-Step Solution

Answer

Identity is verified: \( ad - bc = -(cb - da) \).

Step 1: Expand the Original Determinant

The given determinant is \( \left| \begin{array}{ll} a & b \ c & d \end{array} \right| \). To expand this 2x2 determinant, use the formula: \( ad - bc \). This is the determinant of the original matrix.

Step 2: Expand the Transposed Determinant

The determinant you need to compare with is \( \left| \begin{array}{ll} c & d \ a & b \end{array} \right| \). Expand this as well using the same formula: \( cb - da \).

Step 3: Simplify Both Determinants

From Step 1, we have \( ad - bc \). From Step 2, we have \( cb - da \). Notice that \( cb - da \) can be rearranged as \( -(da - cb) = -(ad - bc) \).

Step 4: Confirm the Identity

By substituting the simplified forms from Step 3 into the given identity, we get \( ad - bc = -(cb - da) \). As derived earlier, \( cb - da = -(ad - bc) \). This confirms that the identity holds true.

Key Concepts

2x2 Determinant

A 2x2 determinant is a special case of a matrix determinant that involves a matrix with two rows and two columns. It's one of the simplest forms to work with, making it a perfect starting point if you're new to matrices.
To calculate the determinant of a 2x2 matrix, which looks like this:

• Matrix: \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\)

you use the formula \(ad - bc\). Simply multiply the top left element \(a\) with the bottom right \(d\), minus the product of the top right \(b\) and bottom left \(c\).
This calculation gives you a scalar value that helps to determine properties of the matrix, such as whether it has an inverse. If the determinant is zero, for instance, the matrix is singular and does not have an inverse. Knowing how to find the determinant is a fundamental skill in linear algebra and helps in solving linear equations and finding eigenvalues.

Matrix Transposition

Matrix transposition is a simple yet powerful concept in linear algebra. Transposing a matrix essentially means flipping it over its diagonal.
If you have a matrix like this:

• Original: \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\)

After transposing, it looks like:

• Transposed: \(\begin{pmatrix} a & c \ b & d \end{pmatrix}\)

Transposing switches the rows and columns. In our 2x2 example, the element in the top row, second column (\(b\)) swaps places with the element in the bottom row, first column (\(c\)). This property is helpful in many matrix operations, including simplifying expressions and solving complex matrix equations. A transposed matrix retains certain properties of the original matrix, like determinant, but can change the way they are organized, making them easier to manipulate in specific algebraic contexts.

Matrix Identity Verification

Matrix identities often provide ways to check relationships and properties within matrices. For instance, the identity given in the exercise involves verifying that the determinant of a matrix is the negative of the determinant of its transposed matrix.This particular identity can be expressed as:

• \( \left|\begin{array}{ll} a & b \ c & d \end{array}\right| = -\left|\begin{array}{ll} c & d \ a & b \end{array}\right| \)

What this means in practice is that if you compute the determinant of a 2x2 matrix \(ad - bc\), then calculate the determinant of its transpose \(cb - da\), you will find one is the negative of the other.
This confirms the property that a matrix and its transpose share determinants with opposite signs under certain conditions. Understanding identity verification like this helps in uncovering symmetries and invariants, which are essential tools for solving complex problems in linear algebra.