Problem 30 - Intermediate Algebra

Question

Evaluate each determinant. $$ \left|\begin{array}{rrr} 1 & 1 & 2 \\ 2 & 1 & -2 \\ 3 & 1 & 3 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is -9.

Step 1: Identify the Determinant Formula

The determinant of a 3x3 matrix \(A\) can be calculated using the formula: \[det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]for a matrix:\[\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]

Step 2: Assign Matrix Values

Given the matrix: \[\begin{bmatrix} 1 & 1 & 2 \ 2 & 1 & -2 \ 3 & 1 & 3 \end{bmatrix}\]assign: \(a = 1\), \(b = 1\), \(c = 2\), \(d = 2\), \(e = 1\), \(f = -2\), \(g = 3\), \(h = 1\), \(i = 3\).

Step 3: Calculate Each Component

Calculate each part of the formula:- \(ei - fh = (1 \times 3) - (1 \times -2) = 3 + 2 = 5\)- \(di - fg = (2 \times 3) - (-2 \times 3) = 6 + 6 = 12\)- \(dh - eg = (2 \times 1) - (1 \times 3) = 2 - 3 = -1\)

Step 4: Substitute and Calculate the Determinant

Substitute the values calculated in Step 3 back into the determinant formula:\[ det(A) = 1 \times 5 - 1 \times 12 + 2 \times (-1)\]Simplify:\[ = 5 - 12 - 2 = -9\]

Key Concepts

Understanding the 3x3 Matrix

A 3x3 matrix is a type of rectangular array consisting of three rows and three columns, containing nine individual elements. Each element can be represented by a unique position in the matrix, denoted typically as \(a_{ij}\), where \(i\) is the row number and \(j\) is the column number. This type of matrix is commonly used in various fields like physics, computer graphics, and statistics, as it provides a compact way to represent linear transformations and various operations.

A matrix is a crucial concept in matrix algebra and often used to solve systems of equations. The matrix in our exercise is:

• First row: 1, 1, 2
• Second row: 2, 1, -2
• Third row: 3, 1, 3

The position of each element within this matrix plays a significant role in calculating the determinant, which we'll discuss next.

Decoding the Determinant Formula

The determinant formula for a 3x3 matrix might initially seem complex, but breaking it down into parts makes it much more approachable. The determinant helps to find out if the matrix is invertible and gives important properties about the system of equations represented by the matrix.

The determinant of a 3x3 matrix \(A\), with elements arranged as:\[A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\] is calculated by the formula:

• Compute: \(ei - fh\)
• Compute: \(di - fg\)
• Compute: \(dh - eg\)

Finally, combine these to obtain the determinant:\[det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]In our example, substituting the given matrix values into this formula will yield the determinant of the matrix.

Basics of Matrix Algebra

Matrix algebra is an essential branch of mathematics that involves the study of matrices and their operations. These operations include addition, subtraction, multiplication, and finding determinants. Each of these has specific rules and properties.

In matrix algebra, addition and subtraction are performed element-wise, while multiplication involves rows and columns according to specific techniques. Unlike numbers, matrix multiplication is not commutative; this means \(AB eq BA\).

Finding the determinant, as we did in our previous section, is a crucial part of understanding matrix properties. It provides information on the invertibility of a matrix (a matrix with a non-zero determinant is invertible). Matrix algebra not only underpins practical applications, such as solving linear equations, but also theoretical concepts in higher mathematics.