Problem 28 - Precalculus: Mathematics for Calculus

Question

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$\left[\begin{array}{rrrr} 1 & 2 & 0 & 2 \\ 3 & -4 & 0 & 4 \\ 0 & 1 & 6 & 0 \\ 1 & 0 & 2 & 0 \end{array}\right]$$

Step-by-Step Solution

Answer

The matrix has a non-zero determinant, so it is invertible.

Step 1: Identify the Matrix

We are given a 4x4 matrix, written as: \[\begin{bmatrix}1 & 2 & 0 & 2 \3 & -4 & 0 & 4 \0 & 1 & 6 & 0 \1 & 0 & 2 & 0 \\end{bmatrix}\] Our task is to find its determinant to determine if the matrix is invertible.

Step 2: Determine if the Matrix has Full Rank

Check if the matrix is of full rank by ensuring that one of its determinants of a 3x3 minor is non-zero. Compute a minor taking rows 1, 2, 3 and columns 1, 2, 3 for simplification.

Step 3: Set Up the 3x3 Minor

Select the first three rows and first three columns to form a 3x3 submatrix: \[\begin{vmatrix}1 & 2 & 0 \3 & -4 & 0 \0 & 1 & 6 \\end{vmatrix}\] as the minor to check.

Step 4: Calculate the Determinant of the 3x3 Minor

Use the rule for calculating the determinant of a 3x3 matrix, which is: \\[\det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)\]For our minor, sub in the values: \\[1((-4)(6) - (0)(1)) - 2((3)(6) - (0)(0)) + 0((3)(1) - (-4)(0))\]\[= 1((-24)) - 2(18) + 0(3)\]\[-24 - 36 = -60\] The determinant of the minor is \(-60\), which is not zero.

Step 5: Conclusion on Determinant and Invertibility

Since the minor determinant is \(-60\), the determinant of the full 4x4 matrix is not zero. Therefore, the matrix is of full rank and is invertible.

Key Concepts

Invertibility

Understanding whether a matrix is invertible is essential in linear algebra. Invertibility means that we can find another matrix, known as the inverse, which, when multiplied by the original matrix, yields the identity matrix.
For a matrix to be invertible:

• The determinant of the matrix must not be zero.
• The matrix must be of full rank, meaning all its rows and columns are linearly independent.

Since the determinant of a matrix is a crucial factor in determining invertibility, calculating it helps us decide if we can find an inverse. For the matrix in our exercise, we checked a 3x3 minor and found it had a non-zero determinant. This affirms that the complete matrix has full rank and, therefore, it is invertible.
It's always a backdoor check: no matter the matrix's size, if any determinant of a square minor is non-zero, the matrix is likely invertible as it has full rank.

4x4 Matrix

When working with matrices, it's common to encounter a 4x4 matrix. These matrices have four rows and four columns and can represent complex systems in mathematics and engineering.
To determine properties like invertibility, we often need to calculate their determinant.
Calculating the determinant of a 4x4 matrix can be intricate, as it involves breaking it down into smaller 3x3 minors.

• Select any row or column and form a series of 3x3 matrices by omitting that row and column for each element.
• Compute each minor's determinant, then sum these up using cofactors.

Though solving for a determinant might seem complex, recognizing patterns can simplify the work. In practice, use smaller matrices within the 4x4 matrix to ease calculations.

Matrix Rank

Matrix rank is an indicator of the dimensions of the vector space spanned by its rows or columns. It's a measure of the matrix's "non-degeneracy."
A matrix is said to have full rank if its rank equals the number of rows or columns. For instance, a 4x4 matrix must have a rank of 4 to be considered full rank, indicating that none of its rows are linear combinations of others.

• A full-rank matrix is crucial: it signifies that the matrix is invertible.
• When determining invertibility, checking the matrix's rank first can save time by indicating potential issues.

In example exercises, finding a non-zero determinant of any minor assures us the matrix is of full rank, confirming invertibility.
Consider the matrix size to determine rank; in large matrices, this can significantly impact computation time and strategy.