Problem 16 - Algebra and Trigonometry

Question

Find the inverse of the matrix if it exists. \(\left[\begin{array}{rrr}{5} & {7} & {4} \\ {3} & {-1} & {3} \\ {6} & {7} & {5}\end{array}\right]\)

Step-by-Step Solution

Answer

The inverse of the matrix does not exist because its determinant is zero.

Step 1: Determine if the Inverse Exists

To check if the inverse of a matrix exists, first calculate the determinant of the matrix. For a 3x3 matrix \(\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \ \end{bmatrix}\), the determinant \( \det(A) \) is given by:\[ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]Applying this formula to the given matrix:\[ \det\left(\begin{bmatrix} 5 & 7 & 4 \ 3 & -1 & 3 \ 6 & 7 & 5 \end{bmatrix}\right) = 5((-1)(5) - 3(7)) - 7((3)(5) - (3)(6)) + 4((3)(7) - (-1)(6)) \]Calculating each term, we find the determinant equals 0, hence the inverse does not exist as the determinant is zero.

Step 2: Conclusion on the Inverse

Since the determinant of the matrix is zero, the matrix is singular. A singular matrix does not have an inverse.

Key Concepts

Determinant Calculation

The determinant of a matrix is a special number that can be calculated from its elements. When dealing with a 3x3 matrix, the process involves a specific formula. Consider a matrix like this:\[\begin{bmatrix} a & b & c \d & e & f \g & h & i \\end{bmatrix}\]To find its determinant, you apply the formula:\[\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]This formula helps by breaking down the large matrix into smaller, more manageable pieces, making the calculation easier.
By finding the determinant, you can understand important properties about the matrix, like whether it has an inverse. Here's why the calculation is important:

• If the determinant is zero, the matrix has no inverse.
• If the determinant is nonzero, the matrix does have an inverse.

In our example, after applying this formula to the given matrix, the determinant was found to be zero, which has profound implications on its invertibility.

Singular Matrix

A matrix is termed "singular" if its determinant equals zero. This terminology is not just a name but speaks to an important property of the matrix. When a matrix is singular, it means that it does not have an inverse. Without an inverse, certain mathematical operations cannot be performed. Understanding singular matrices involves:

• Recognizing that they do not have a unique solution in systems of equations.
• Knowing that operations relying on inverses, like solving matrix equations, are not possible.

Singular matrices often arise in contexts where transformations or changes are constrained.
In simple terms, think of it as a door that won't open because something is blocking it. In mathematics, that "block" is the zero determinant.
Interestingly, in real-world applications, singular matrices can indicate dependent variables or tied conditions, making their detection important in analysis.

3x3 Matrix

A 3x3 matrix is a crucial mathematical concept with three rows and three columns. This structure makes it manageable yet rich enough for complex calculations, like finding determinants or performing matrix multiplication. Key features of 3x3 matrices include:

• Being used frequently in 3D space problems, such as rotations and transformations.
• Serving as a building block in systems of linear equations.
• Providing insights into more complex structures by allowing the calculation of eigenvalues and eigenvectors.

The size allows for a perfect balance: small enough to handle without calculators for practice, yet complex enough to introduce significant concepts in linear algebra. In practical contexts, a 3x3 matrix can represent data or transformations. Think of 3D graphics; each transformation of an object can be represented by multiplying it with a 3x3 matrix, altering its position or orientation in space.