Problem 24 - Algebra and Trigonometry

Question

Evaluate each determinant. $$\left|\begin{array}{rrr}{4} & {0} & {0} \\\\{3} & {-1} & {4} \\\\{2} & {-3} & {5}\end{array}\right|$$

Step-by-Step Solution

Answer

The determinant of the given matrix is 28.

Step 1: Identify the matrix

Identify the given matrix, which is 3x3. Here it is: \[\begin{{bmatrix}}4 & 0 & 0 \\3 & -1 & 4 \\2 & -3 & 5 \end{{bmatrix}}\]

Step 2: Apply the formula for a 3x3 determinant

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

Step 3: Substitute the values into the formula

Now substitute the corresponding elements from the given matrix into the formula\[{= 4(-1*5 - -3*4) - 0(3*5 - 2*4) + 0(3*-3 - 2*-1)}\]

Step 4: Simplify the Expression

Simplify the result \[{= 4(-5 - -12) = 4*7 = 28}\]

Key Concepts

Determinant Formula

To calculate the determinant of a 3x3 matrix, such as the example matrix \[\begin{bmatrix} 4 & 0 & 0 \ 3 & -1 & 4 \ 2 & -3 & 5 \end{bmatrix}\],we need to use the determinant formula for a 3x3 matrix. This formula allows us to calculate the determinant, which is a single number representing certain properties of the matrix.
The formula for a 3x3 matrix \[A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]is:\[det(A) = a(ei−fh)−b(di−fg)+c(dh−eg)\].
This equation might look complex at first, but it involves a combination of multiplication and subtraction operations among the elements of the matrix. Each term in the formula corresponds to a 2x2 determinant of a submatrix formed by excluding the row and column of each matrix element \(a, b, \) and \(c\) respectively. This method is known as cofactor expansion, specifically expanding along the first row.

Matrix Evaluation

Matrix evaluation is the process of performing operations on matrices to derive meaningful results, such as calculating the determinant. In our given matrix, we have a top row that simplifies the calculation because it includes zeros:
\[\begin{bmatrix} 4 & 0 & 0 \ 3 & -1 & 4 \ 2 & -3 & 5 \end{bmatrix}\].
When applying the determinant formula to this matrix, the zeroes in the top row significantly reduce the complexity. Since these zeroes multiply with entire expressions involving submatrices, they simplify to zero in those terms.
Here's what we need to do:

• Compute minor matrices for some matrix elements (exploit zeroes).
• Apply the given formula.
• Simplify step-by-step.

Specifically, the middle and right terms will not affect the determinant because they multiply by zero:\(0(3*5 - 2*4)\) and \(0(3*-3 - 2*-1)\). Thus, we only need to evaluate the term influenced by "4":\[4(-1*5 - (-3)*4)\].By executing these operations accurately, matrix evaluation enables solving mathematical expressions linked with matrices.

Algebra Concepts

Understanding how to work with determinants connects closely with several key algebra concepts. These ideas help in comprehending both the properties of determinants and the steps involved in their calculation.
One important concept is the idea of identity and zero. In matrices, zero is a powerful tool in calculations because it simplifies terms in equations, especially when it appears in rows or columns like in our 3x3 matrix.

• This makes calculations easier, often cutting down on computational steps.

Another concept is the distributive property inherent in algebra. When applying the determinant formula, we distribute calculations over the components of the first row, taking advantage of arithmetic operations: multiplication and addition.
Additionally, adjacency in this calculation relates to algebra. By breaking parts of the matrix into smaller components (like 2x2 sections), we apply simpler algebraic manipulations to these smaller parts. This iterative approach follows the cofactor expansion seen in algebraic structures. Equipped with these algebra concepts, you gain a deeper appreciation for how matrices work and how linear algebra builds from elementary algebra principles to manage complex matrix operations.