Problem 21 - Precalculus: Mathematics for Calculus

Question

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$\left[\begin{array}{rrr} 2 & 1 & 0 \\ 0 & -2 & 4 \\ 0 & 1 & -3 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 4; the matrix is invertible.

Step 1: Understanding the Determinant

A matrix has an inverse if its determinant is non-zero. For a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \), the determinant is calculated as \( \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \).

Step 2: Identifying Matrix Elements

The given matrix is \( \begin{bmatrix} 2 & 1 & 0 \ 0 & -2 & 4 \ 0 & 1 & -3 \end{bmatrix} \). Identify the elements: \( a=2, b=1, c=0 \), \( d=0, e=-2, f=4 \), \( g=0, h=1, i=-3 \).

Step 3: Calculate the Cofactors

Calculate the cofactors needed:* \( ei - fh = (-2)(-3) - (4)(1) = 6 - 4 = 2 \)* \( di - fg = (0)(-3) - (4)(0) = 0 - 0 = 0 \)* \( dh - eg = (0)(1) - (-2)(0) = 0 - 0 = 0 \)

Step 4: Compute the Determinant

The determinant of the matrix is given by:\[ \text{det}(A) = 2(2) - 1(0) + 0(0) = 4 \].

Step 5: Determine Invertibility

Since the determinant \( \text{det}(A) = 4 \) is non-zero, the matrix is invertible.

Key Concepts

Matrix Inverse

In mathematics, finding the inverse of a matrix is akin to solving a puzzle. An inverse matrix is significant because, when multiplied with the original matrix, it gives the identity matrix. This concept is especially useful in solving systems of linear equations. However, not all matrices have an inverse. A matrix can only be inverted when its determinant is non-zero.

For a matrix \( A \), if \( A^{-1} \) exists, it satisfies the equation:

• \( AA^{-1} = A^{-1}A = I \)

where \( I \) is the identity matrix, essentially a matrix that doesn't change other matrices when multiplied by them. In real-world applications, inverses are used in everything from computer graphics to cryptography. Remember, finding the actual inverse can be quite complex and involves finding determinants and using methods such as Gaussian elimination or the adjoint method.

Cofactor Expansion

Understanding the concept of cofactor expansion is crucial in determining the determinant of a matrix. Cofactor expansion, also known as Laplace's expansion, allows you to expand a determinant along any row or column of the matrix, providing flexibility in computation.

Here's how it works:

• Select a row or column to expand along. Choose whichever has the most zeros for easier computation.
• For each element in this row or column, calculate its minor, which is the determinant of the submatrix that remains after removing the element's row and column.
• Then, determine the cofactor by applying a checkerboard pattern of signs to the minors, switching between positive and negative.
• Sum up the products of each element, its cofactor, and the matrix expands.

In a 3x3 matrix, using cofactor expansion helps in simplifying the computation of determinants by breaking down a potentially complex procedure into manageable parts.

Invertibility of Matrices

The concept of matrix invertibility is fundamental in linear algebra. Matrix invertibility refers to whether a matrix has an inverse. A crucial criterion for this is the determinant of the matrix. If the determinant is zero, the matrix is singular and does not have an inverse.

Here's why the determinant being non-zero matters:

• It is a test for linear independence of columns or rows: a zero determinant indicates linear dependence, which means no inverse exists.
• A non-zero determinant implies that row echelon form or reduced row echelon form will eventually yield an identity matrix during calculations, granting the matrix an inverse.

In our exercise example, by computing the determinant as 4, a non-zero value, we determined that the matrix is invertible. In real-life applications, invertible matrices are especially significant in computations involving transformations and in solving linear systems reliably.