Are you struggling with matrix algebra assignment questions? You're not alone. Many students find this topic challenging, but fear not! In this blog, we'll delve into a tough matrix algebra question typically tested in universities. We'll break it down, explain the concepts, and provide a step-by-step guide to answering it. So, let's dive in!

The Question:
Consider a square matrix A with dimensions =n × n. Prove that if A is invertible, then its determinant is nonzero.

Concept Explanation:
To understand this question, we need to grasp the concept of invertibility and determinants in matrix algebra.

1. Invertibility:
A square matrix A is invertible (or nonsingular) if there exists another matrix B such that AB=BA=I, where I is the identity matrix. In simpler terms, if you can find a matrix that, when multiplied with A, gives the identity matrix, then A is invertible.

2. Determinants:
The determinant of a square matrix is a scalar value that can be computed from its elements. It provides important information about the matrix, such as whether it is invertible. For a square matrix A, denoted as ∣A∣, if A∣≠0∣, then A is invertible.

Step-by-Step Guide:
Now, let's outline how to prove that if A is invertible, then its determinant is nonzero.

1. Assume A is invertible:
Start by assuming that A is invertible, meaning there exists a matrix B such that AB=BA=I.

2. Use the definition of determinants:
Recall that the determinant of a matrix can be expressed using its elements. For a square matrix A, ∣A∣ can be computed using various methods such as cofactor expansion or Gaussian elimination.

3. Show that ∣A∣≠0∣
Since A is invertible, there exists a matrix B such that AB=I. Now, multiply both sides of this equation by ∣A∣. This yields ∣AB∣=∣I∣.

4. Apply properties of determinants:
The determinant of the product of two matrices is equal to the product of their determinants. Thus, ∣AB∣=∣A∣∣B∣. Also, the determinant of the identity matrix is 1.

5. Conclude:
Combining the above steps, we have ∣A∣∣B∣=∣A∣⋅1. Since =1.21em�A is invertible, ∣A∣≠0∣.

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