Problem 40 - Differential Equations and Linear Algebra

Question

If \(A, B, C\) are \(n \times n\) matrices satisfying \(B A=I_{n}\) and \(C A=I_{n},\) does it follow that \(B=C ?\) Justify your answer.

Step-by-Step Solution

Answer

We cannot conclude that \(B = C\) based on the given information, as we cannot guarantee the existence of the inverse of matrix \({(A^2)}^{-1}\) which could help us derive a relationship between matrices \(B\) and \(C\).

Step 1: Use the matrix product properties

We know from the problem that \((B A) = I_{n}\) and \((C A) = I_{n}\). First, multiply both equations by matrix \(A\) from the right. $$B A A = I_n A$$ $$C A A = I_n A$$

Step 2: Use the associativity property

Matrix multiplication is associative, which means that \((A B) C = A (B C)\) for any matrices \(A\), \(B\), and \(C\). So, we can rewrite the equations in Step 1 by applying the associativity property: $$B (A A) = (I_n) A$$ $$C (A A) = (I_n) A$$

Step 3: Simplify equations

We can simplify these equations by noticing that \(A A = A^2\) and \(I_n A = A\). Therefore, $$B (A^2) = A$$ $$C (A^2) = A$$

Step 4: Determine if B = C

Since both sides of the equation are equal, $$B (A^2) = C (A^2)$$ Now, we need to determine if we can divide both sides by \(A^2\). However, matrix division does not exist in a general sense, and we cannot guarantee that the inverse of matrix \({(A^2)}^{-1}\) exists. It means we cannot simplify the above equation further, in order to show that \(B = C\). Therefore, we cannot conclude that \(B = C\) based on the given information.

Key Concepts

Associative Property of Matrices

Understanding the associative property of matrices is crucial when dealing with matrix multiplication. This property dictates that the order in which matrices are grouped during multiplication does not change the result. In other words, for any matrices A, B, and C, the equation (AB)C = A(BC) will always hold true.

This property is particularly helpful when we need to simplify complex matrix expressions, as it allows us to rearrange the groupings of matrices without affecting the outcome. This flexibility can greatly ease the process of manipulating matrices to reach a solution or to prove a concept, like comparing the product of multiple matrices to determine if two variables are equal in a given context.

Application in Exercises

When faced with an equation like B(AA) = (In)A, we can safely group A with itself to form A2 due to the associative property, without any concern that this might alter the result. This step is essential in proving more complex properties or in understanding why certain algebraic manipulations are used in exercises.

Matrix Inverses

The concept of matrix inverses is akin to division in conventional algebra. For a given square matrix A, its inverse is denoted as A-1 and it satisfies the equation AA-1 = A-1A = In, where In is the identity matrix of the same dimension.

This property is foundational because the inverse of a matrix can be used to solve systems of linear equations, to find matrix division equivalents, and to perform transformations in geometry. It is worth noting, however, that not all matrices have an inverse. A matrix must be non-singular, meaning it has a non-zero determinant, to possess an inverse.

Significance in Equations

The existence of a matrix inverse is central to the understanding that we cannot simply 'divide' by a matrix in the same way we divide by numbers. In our exercise scenario, since we are unable to confirm the existence of (A2)-1, we cannot proceed to 'cancel out' as we might with numerical division. This limitation is a key aspect of matrix manipulation and underpins why we cannot conclude B = C in the provided exercise without additional information.

Identity Matrix

The identity matrix, often denoted as I_n for an n x n matrix, is the matrix equivalent of the number 1 in arithmetic. It is a special kind of diagonal matrix where all the elements on the main diagonal are 1, and the rest are 0. When you multiply any matrix A by the identity matrix, you get the same matrix A back: AI_n = I_nA = A.

This property makes the identity matrix a neutral element in matrix multiplication, just as multiplying a number by 1 does not change its value. When solving equations or transforming matrices, the concept of the identity matrix is essential because it preserves the original matrix.

Role in Matrix Equations

In our exercise context, multiplying by the identity matrix I_nA results simply in A, reinforcing the inherent concept that the identity matrix does not alter other matrices it multiplies. It also plays a role in understanding that when we equate two products of matrices to an identity matrix, as in BA = CA = In, it signifies that both B and C are uniquely related to A as potential inverses, a relationship that hinges on the properties of the identity matrix itself.