Problem 57 - Precalculus

Question

find \(A^{-1},\) if possible. $$A=\left[\begin{array}{rrr}1 & 0 & 1 \\\0 & 1 & 1 \\\1 & -1 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

The matrix \( A \) is not invertible because its determinant is 0.

Step 1: Expression

find \(A^{-1},\) if possible.
$$A=\left[\begin{array}{rrr}1 & 0 & 1 \\\0 & 1 & 1 \\\1 & -1 &
0\end{array}\right]$$

Step 2: Apply Rules

Apply appropriate mathematical rules step by step.

Step 3: Result

The matrix \( A \) is not invertible because its determinant is 0.

Key Concepts

Determinant of a Matrix

The determinant is a special number that can be calculated from a square matrix. It plays a vital role in determining if a matrix has an inverse. For a 3x3 matrix, the determinant calculation can seem complex at first.
If you have a matrix

• \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \)

The determinant \( \text{det}(A) \) is found using:

• \( a(ei - fh) - b(di - fg) + c(dh - eg) \).

This expression shows a combination of products from the elements of the matrix. Understanding this formula helps in determining the magnitude of a matrix.
When the determinant is zero, it indicates that the matrix does not span the full space and is singular, meaning it lacks an inverse.
Hence, checking the determinant is a crucial first step when dealing with matrices if you need to find an inverse.

Invertibility of Matrices

A matrix is said to be invertible if there exists another matrix that, when multiplied with it, results in the identity matrix. This concept is equivalent to the idea of multiplicative inverses with numbers, like how 1/2 is the inverse of 2.
For a matrix to be invertible, especially in the context of a square matrix, its determinant must be non-zero.
This ensures that the matrix can completely span its vector space.
In simple terms, invertibility means you can 'undo' the transformation that the matrix represents.

• If \( \text{det}(A) eq 0 \), the matrix is invertible.
• If \( \text{det}(A) = 0 \), the matrix is not invertible.

In our exercise, since the determinant is zero, the matrix is not invertible. This is significant when solving systems of linear equations, as a non-invertible matrix indicates there might be no unique solution or infinitely many solutions.

Matrix Algebra

Matrix algebra involves rules and operations for manipulating matrices. It includes operations such as addition, subtraction, multiplication, and finding inverses.
Each operation has its own rules and characteristics, especially multiplication.

• Matrix multiplication is not commutative, meaning \( AB eq BA \) in general.
• The identity matrix plays a role similar to the number 1 in regular multiplication.

When a matrix is multiplied by its inverse, it results in the identity matrix. However, this is only possible if the matrix is invertible.
Understanding these operations helps in solving linear equations, transforming data, and even in more advanced applications in computer science and engineering.
This foundation of matrix algebra is essential in fields such as machine learning, physics, and more. For instance, understanding how to calculate determinants and find inverses are critical skills in matrix algebra.