Problem 19 - Advanced Engineering Mathematics

Question

Find the inverse of the given matrix or show that no inverse exists. $$ \left(\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right) $$

Step-by-Step Solution

Answer

The matrix is singular and has no inverse due to a zero determinant.

Step 1: Evaluate the Determinant

To find the inverse of a matrix, the determinant must be non-zero. The determinant of a 3x3 matrix \(A\), \( \det(A) \), is calculated as follows:\[ \det\left(\begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array}\right) = a(ei - fh) - b(di - fg) + c(dh - eg) \]For the given matrix \(\left(\begin{array}{ccc} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{array}\right)\):\[ \det = 1(5 \times 9 - 6 \times 8) - 2(4 \times 9 - 6 \times 7) + 3(4 \times 8 - 5 \times 7) \]\[ = 1(45 - 48) - 2(36 - 42) + 3(32 - 35) \]\[ = 1(-3) - 2(-6) + 3(-3) \]\[ = -3 + 12 - 9 \]\[ = 0 \] The determinant is zero.

Step 2: Conclusion on Inversibility

Since the determinant is zero, the matrix does not have an inverse. For a matrix to have an inverse, the determinant must be non-zero. Therefore, the given matrix is singular and does not have an inverse.

Key Concepts

Determinants

Determinants are a critical concept when determining if a matrix has an inverse. In simple terms, the determinant is a special number that can be calculated from a square matrix. Knowing how to compute the determinant of a matrix helps you understand its properties.

For a 3x3 matrix, the determinant is calculated using a combination of subtraction and multiplication, according to a specific formula. The determinant of a matrix \[\left(\begin{array}{ccc} a & b & c \ d & e & f \ g & h & i \end{array}\right)\]is given by:

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

A determinant's role is pivotal as it indicates whether a matrix is invertible or not. If the determinant equals zero, the matrix won't have an inverse. In our example, the calculation resulted in a determinant of zero, indicating no inverse exists.

Singular Matrices

When we talk about singular matrices, it's important to understand that they are the matrices which do not have an inverse. A matrix becomes singular when its determinant is zero. This characteristic makes singular matrices a special type of square matrix.

Since the calculation of the determinant is an essential step, you should always find whether it's zero or not. If the determinant is zero as with our matrix, it suggests that there's no solution to the equation \(Ax = b\) where \(A\) is the matrix in question. A singular matrix represents a system of equations that is either dependent or has no solution at all.

In essence, identifying singular matrices is crucial in understanding the limitations they impose on mathematical operations, especially when determining if an inverse exists.

Non-invertible Matrices

Non-invertible matrices, sometimes called "degenerate" matrices, are those that cannot be inverted. The reason lies in the fact that their determinant is zero. Without a non-zero determinant, you cannot move forward with finding an inverse because this number is required in the formula to calculate one.

For any matrix \(A\), if its determinant \(\det(A) = 0\), there is no inverse. This is why such matrices are also singular. They fundamentally lack the property needed to find an inverse matrix, and attempting to invert them would lead to undefined operations or unsolvable scenarios in systems of equations.

Non-invertible matrices often appear in linear algebra problems and recognizing them allows for proper analysis of solutions or potential issues within a given mathematical context. Understanding what makes a matrix non-invertible represents a key component in mastering the subject of matrices.