Problem 28 - College Algebra and Calculus: An Applied Approach

Question

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{lll} 1 & 0 & 0 \\ 3 & 0 & 0 \\ 2 & 5 & 5 \end{array}\right] $$

Step-by-Step Solution

Answer

The given matrix does not have an inverse because its determinant is equal to zero.

Step 1: Checking if the matrix has an inverse

A matrix has an inverse only if it's a square matrix and its determinant is non-zero. The given matrix is a square matrix. So, calculate the determinant of the given matrix. The determinant (Det) of a 3x3 matrix: \[Det(A)=a(ei−fh)−b(di−fg)+c(dh−eg)\]By substituting the values of the given matrix into the determinant equation, \( Det = 1*(0*5 - 0*0) - 0*(3*5 - 0*0) + 0*(3*0 - 2*0) = 0 \). The determinant is zero, therefore, the matrix does not have an inverse.

Step 2: Conclusion

As the determinant of the matrix is zero, we concluded that our matrix does not possess an inverse. Only square matrices with non-zero determinants can have an inverse. Since our matrix does not fulfill that criteria, it drastically simplifies our problem, as there are no further steps necessary.

Key Concepts

Determinant Calculation

Calculating the determinant is a critical step in determining if a matrix has an inverse. The determinant is a special number associated with square matrices, and it provides important properties about the matrix. For a 3x3 matrix, the determinant can be found using the following formula:\[Det(A) = a(ei−fh)−b(di−fg)+c(dh−eg)\]This formula involves multiplying and subtracting elements from the matrix systematically.
In essence, for any 3x3 matrix, you'll need to:

• Take an element from the top row,
• Multiply it by a 2x2 determinant formed by deleting the row and column of the element,
• Repeat the process for each element in the top row, adjusting signs accordingly.

A matrix has no inverse if its determinant is zero, because it indicates that the matrix is singular, meaning it cannot map to an inverse transformation.

Square Matrices

A square matrix is a matrix with the same number of rows and columns. They are special because only these matrices can potentially have an inverse. The characteristics of square matrices include:

• Consistent dimensions, such as 2x2, 3x3, etc.,
• A main diagonal from top left to bottom right.

Square matrices come up in various mathematical contexts, providing the foundation for many operations, including determinants and inverses.
When dealing with matrices, identifying whether you're working with a square matrix is crucial because only these types can offer solutions via the inverse, provided their determinant is non-zero.

Inverse Matrix

An inverse matrix is essentially a matrix that reverses the effect of the original matrix. To have an inverse, a matrix must be square and its determinant must be non-zero. The process of finding an inverse involves several steps, primarily:

• Calculate the determinant to ensure the matrix can have an inverse,
• If non-zero, apply techniques such as the adjugate method, paired with dividing by the determinant.

If a matrix fails the determinant test, which means it is zero, then the matrix is said to be singular and automatically lacks an inverse.
This property of an inverse matrix is used when solving systems of equations, where finding the inverse allows one to find unique solutions to the system by matrix operations.