Is a 3×3 matrix always invertible?

Answered by Jarrod Smith

Is a 3×3 matrix always invertible?

To answer this question, let’s first understand what it means for a matrix to be invertible. A matrix A is invertible if there exists another matrix B such that the product of A and B is the identity matrix. In other words, if AB = BA = I, where I is the identity matrix.

In the case of a 3×3 matrix, we can determine if it is invertible by calculating its determinant. The determinant of a matrix is a scalar value that provides information about the matrix’s properties. For a 3×3 matrix A, the determinant is denoted as det(A) or |A|.

If the determinant of a 3×3 matrix is non-zero, then the matrix is invertible. However, if the determinant is zero, then the matrix is not invertible. This is because the determinant plays a crucial role in finding the inverse of a matrix. If the determinant is zero, it implies that the matrix has linearly dependent rows or columns, making it impossible to find a unique inverse.

Now, let’s consider the given question: Is a 3×3 matrix always invertible? The answer is no. A 3×3 matrix is not always invertible. It is only invertible if its determinant is non-zero.

To further illustrate this point, let’s take a look at a couple of examples.

Example 1:
Consider the following 3×3 matrix:
A = [[1, 2, 3],
[4, 5, 6],
[7, 8, 9]]

To find the determinant of A, we can use various methods such as cofactor expansion or row reduction. Using the cofactor expansion method, we have:
|A| = 1*(5*9 – 6*8) – 2*(4*9 – 6*7) + 3*(4*8 – 5*7)
= 1*(45 – 48) – 2*(36 – 42) + 3*(32 – 35)
= -3 + 12 – 9
= 0

Since the determinant of A is zero, we can conclude that this matrix is not invertible.

Example 2:
Now, let’s consider a different 3×3 matrix:
B = [[1, 2, 3],
[0, 1, 4],
[0, 0, 1]]

Again, we can calculate the determinant using the cofactor expansion method:
|B| = 1*(1*1 – 4*0) – 2*(0*1 – 4*0) + 3*(0*0 – 1*0)
= 1 – 0 + 0
= 1

In this case, the determinant of B is non-zero, indicating that the matrix B is invertible.

From these examples, we can see that a 3×3 matrix is not always invertible. The invertibility of a 3×3 matrix depends on its determinant. If the determinant is non-zero, the matrix is invertible; otherwise, it is not.

A 3×3 matrix is not always invertible. Its invertibility is determined by the value of its determinant.