Problem 10 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the determinant of the matrix. $$\left[\begin{array}{lr}4 & -3 \\\0 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the given matrix is 0

Step 1: Identifying the elements of the matrix

In the given matrix \( \left[ \begin{array}{cc} 4 & -3 \ 0 & 0 \end{array} \right] \), \( a = 4 \), \( b = -3 \), \( c = 0 \), and \( d = 0 \).

Step 2: Applying the determinant formula

Substitute the identified matrix elements into the determinant formula \(ad - bc\). Thus, the determinant becomes \(4*0 - 0*(-3) = 0\).

Key Concepts

Matrix

A matrix is an array of numbers, or 'elements', arranged in rows and columns, and is an essential concept in linear algebra. Matrices are often denoted by uppercase letters and the numbers within them are called its entries or elements. They can represent systems of linear equations, transformations, and much more. For example, the matrix given in the exercise is
\[\left[\begin{array}{cc}4 & -3 \0 & 0 \end{array}\right]\]
and is composed of two rows and two columns. The first row consists of the numbers 4 and -3, while the second row consists only of zeroes.

Determinant Formula

The determinant is a special number that can be calculated from a square matrix. It provides important information about the matrix, such as whether the matrix has an inverse, and the volume distortion during a linear transformation represented by the matrix.

Determinant of a 2x2 Matrix

For a 2x2 matrix like in the exercise, the determinant formula is defined as
\( ad - bc \)
where \( a, b, c, \) and \( d \) are elements of the matrix positioned as follows:
\[\left[\begin{array}{cc}a & b \c & d \end{array}\right]\]
Essentially, you multiply the diagonal elements and subtract the product of the off-diagonal elements to find the determinant. As seen in the given exercise, after substituting the matrix elements, the determinant calculation simplifies to zero.

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. In understanding determinants, algebraic principles are used to manipulate the expressions and solve the matrix equations. When substituting the variables with actual numbers, as in the step-by-step solution for the determinant, simple algebra simplifies the matrix to a single numerical value, which, in the exercise provided, turned out to be zero, thus demonstrating a fundamental application of algebra in linear algebra problems.