Problem 22 - Precalculus with Limits

Question

In Exercises 13-24, find the inverse of the matrix (if it exists). \(\left[ \begin{array}{r} 1 & 0 & 0 \\ 3 & 0 & 0 \\ 2 & 5 & 5\end{array} \right]\)

Step-by-Step Solution

Answer

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

Step 1 - Check if the Matrix is Invertible

The given matrix is invertible if its determinant is not equal to zero. A square matrix has an inverse only when the determinant is not equal to zero. So, calculate the determinant of the 3x3 matrix. The determinant of a 3x3 matrix \( \left[ \begin{array}{r} a & b & c \ d & e & f \ g & h & i\end{array}\right] \) is given by \(a(ei−fh)−b(di−fg)+c(dh−eg)\). Substituting the values from the given matrix, the determinant will be \(1(0*5-0*0)-0(3*5-0*2)+0*(3*5-2*0)=1*0-0+0=0\). The determinant of the given matrix is zero, which means the given matrix is not invertible.

Step 2 - Finding the Inverse

Since the determinant of the matrix is zero, the matrix does not have an inverse. The existence of the inverse of a matrix is only if its determinant is not equal to zero.

Key Concepts

Invertible Matrix

An invertible matrix plays a crucial role in linear algebra. It is a matrix that can be reversed, i.e., there exists another matrix that, when multiplied with the original matrix, yields an identity matrix. This concept is fundamental because it allows us to solve systems of linear equations efficiently.

• Criterion for invertibility: A square matrix is invertible if and only if its determinant is not zero.
• Identity matrix: A matrix is invertible if it can be multiplied with another matrix to produce an identity matrix, which is a diagonal matrix with all diagonal elements being 1, and all others being 0.

However, if the determinant equals zero, there is no such matrix that can refund the identity matrix by multiplication; thus, the given matrix is not invertible. This non-invertibility means a system of equations corresponding to this matrix would either have no solution or infinitely many solutions.

3x3 Matrix

A 3x3 matrix is a square matrix with three rows and three columns. These matrices are common in many fields such as physics, computer graphics, and engineering.

• Square matrix: A 3x3 matrix is a type of square matrix, meaning it has an equal number of rows and columns.
• Determinant calculation: Calculating the determinant of a 3x3 matrix involves a specific formula, where you take a combination of products and subtractions to deduce a single scalar.

Understanding 3x3 matrices is essential as they provide a manageable yet complex enough framework for computations, offering deep insights into linear transformations and the geometry of spaces these transformations affect.

Matrix Inverse

The inverse of a matrix is akin to division in numbers, offering a matrix that can 'undo' the effect of a multiplication. For a 3x3 matrix, finding the inverse involves more complex calculations than simply dividing another number.

• Inverse Calculation: If a matrix is invertible, its inverse can be calculated using various methods, such as Gaussian elimination or by applying the adjugate method.
• Importance: Matrix inverses are used in various applications, including solving linear equations, linear programming, and computer graphics.

When a matrix's determinant is zero, it signifies singularity, meaning the matrix does not have an inverse. Recognizing whether a matrix is invertible is vital before attempting inverse calculations, ensuring the matrix yields an accurate inverse when permissible.