Problem 7 - Algebra and Trigonometry

Question

Evaluate each determinant. $$ \left|\begin{array}{cc} {-5} & {-1} \\ {-2} & {-7} \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant of the given matrix is 33.

Step 1: Identify the Elements of the Matrix

The matrix is:\n\n\[\begin{array}{cc} -5 & -1 \ -2 & -7 \end{array}\]\n\nSo, the elements of our matrix are: \(a = -5\), \(b = -1\), \(c = -2\), and \(d = -7\).

Step 2: Apply the Determinant Formula for a 2x2 Matrix

The determinant of a 2x2 matrix is calculated as:\n\n\[det = ad - bc\]\n\nSubstitute the values of a, b, c, and d into the formula. Therefore:\n\n\[det = (-5)(-7) - (-1)(-2)\]

Step 3: Calculate the Determinant

Calculate the values:\n\n\[det = 35 - 2\]\n\nTherefore, the determinant is:\n\n\[det = 33\]

Key Concepts

Determinant of a Matrix

The determinant is a special number that can be calculated from a matrix. It provides important information about the matrix, including whether it has an inverse and the volume of geometrical figures in relation to that matrix. Specifically, a non-zero determinant implies that the matrix is invertible, while a zero determinant suggests that the matrix has no inverse.

To calculate the determinant of a 2x2 matrix, you use a simple formula that involves multiplication and subtraction of the elements of the matrix. The process illustrates the fundamental property of determinants: they change sign when two rows (or columns) are swapped and become zero if two rows (or columns) are identical.

Improving the understanding of evaluating determinants includes recognizing patterns in matrix transformations, such as scaling or rotating figures in the plane, which also affects the determinant value. Thinking of determinants in terms of geometric transformations can help with conceptual understanding.

2x2 Matrix

A 2x2 matrix is a square array consisting of two rows and two columns, usually represented in the form

\begin{center} \[ \begin{array}{cc} a & b \ c & d \end{array} \]
Understanding a 2x2 matrix requires familiarity with notation and the ability to perform basic matrix operations, including determinant evaluation, matrix addition, and multiplication. The structural simplicity of 2x2 matrices makes them ideal for introducing the foundational concepts of higher-dimensional matrices.

Algebraic Operations

Algebraic operations include addition, subtraction, multiplication, and division of numbers and expressions. In the context of matrices, these operations follow specific rules that differ from regular arithmetic. For instance, matrix multiplication is not commutative, meaning the order of multiplication matters immensely.

When working with determinants and matrices, it is essential to apply these operations accurately. Simplification of algebraic expressions is a fundamental skill in this area, as it allows for the computation of matrix properties, such as the determinant. Improving on these algebraic operations will enable students to handle more complex problems involving matrices, including finding eigenvalues and solving systems of equations.