Problem 28 - Precalculus: Functions and Graphs

Question

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

Step-by-Step Solution

Answer

The identity is verified as both expressions simplify to \(ad + af - bc - be\).

Step 1: Calculate the Determinant of the First Matrix

The determinant of \(\begin{pmatrix}a & b \ c & d\end{pmatrix}\) is calculated using the formula for a 2x2 matrix: \(ad - bc\).

Step 2: Calculate the Determinant of the Second Matrix

The determinant of \(\begin{pmatrix}a & b \ e & f\end{pmatrix}\) is also calculated using the 2x2 determinant formula: \(af - be\).

Step 3: Add the Two Determinants

Sum the results from Steps 1 and 2: \((ad - bc) + (af - be) = ad - bc + af - be\).

Step 4: Calculate the Determinant of the Sum Matrix

The determinant of the matrix \(\begin{pmatrix}a & b \ c+e & d+f\end{pmatrix}\) is calculated as: \(a(d+f) - b(c+e) = ad + af - bc - be\).

Step 5: Verify the Identity

Compare the expanded form from Step 3, which is \(ad - bc + af - be\), with the result from Step 4, \(ad + af - bc - be\). Both expressions are identical, thus confirming the identity.

Key Concepts

Exploring the 2x2 Matrix

A 2x2 matrix is a simple yet powerful mathematical structure that consists of two rows and two columns. It is often expressed in the form \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), where \(a, b, c,\) and \(d\) are elements of the matrix. These structures are particularly useful because they can be easily manipulated through operations such as addition, multiplication, and finding determinants. The determinant of a 2x2 matrix is a single number calculated from its elements, providing important insights about the matrix, such as whether it can be inverted. For a matrix \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\), the determinant is given by the formula \(ad - bc\). It's a straightforward computation yet plays a crucial role in matrix algebra.

Essential Matrix Operations

Matrix operations are fundamental mathematical procedures that involve various forms of arithmetic calculations applied to matrices. One key operation is calculating the determinant, especially for 2x2 matrices. Another common operation is the addition of two matrices. When adding matrices, you simply add corresponding elements from each matrix. If given, for example, matrices \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\) and \(\begin{pmatrix} a & b \ e & f \end{pmatrix}\), the result is a matrix with summed corresponding elements. Matrix multiplication is a bit more complex, involving the product of elements and sums. Importantly, practicing these operations enhances your understanding of linear transformations and system solutions.

Understanding Mathematical Proof

Mathematical proof is a logical argument demonstrating the truth of a given statement. In verifying identities like the one in this exercise, you use proofs to show that both sides of an equation produce equivalent results. Proofs can involve calculations, like finding determinants on both sides of the equation given and comparing them. In this exercise, we show that adding determinants of separate matrices results in the same determinant as the combined matrix: from \((ad - bc) + (af - be)\) to \((a(d+f) - b(c+e))\). By examining these calculations step by step, we establish that these expressions are equal, thereby confirming the identity with confidence. This systematic approach is key to developing robust mathematical reasoning skills.