Understanding Properties of Square Matrices

Let's analyze each statement about square matrices A, B, and C of the same order n.

Analyzing Statement 1: AB = AC ⇒ B = C if A is non-singular

This statement explores the property of cancellation in matrix multiplication when one of the matrices is non-singular. A square matrix A is non-singular if and only if its inverse, denoted as \(A^{-1}\), exists. When \(A^{-1}\) exists, we can use it to manipulate matrix equations.

Given the equation: \(AB = AC\)

Since A is non-singular, we can multiply both sides of the equation by \(A^{-1}\) from the left:

\(A^{-1}(AB) = A^{-1}(AC)\)

Using the associative property of matrix multiplication \((XY)Z = X(YZ)\), we can group the terms:

\((A^{-1}A)B = (A^{-1}A)C\)

We know that the product of a matrix and its inverse is the identity matrix, denoted by I (\(A^{-1}A = I\)). The identity matrix behaves like the number 1 in multiplication; multiplying any matrix by the identity matrix (of compatible size) results in the original matrix (\(IB = B\) and \(IC = C\)).

Substituting \(A^{-1}A = I\) into the equation:

\(IB = IC\)

Which simplifies to:

\(B = C\)

Thus, if A is a non-singular matrix and \(AB = AC\), it necessarily follows that \(B = C\). Statement 1 is correct.

Analyzing Statement 2: If BX = CX for every column matrix X having n rows then B = C

This statement checks the implication of a matrix equation holding true for all possible column vectors. We are given that \(BX = CX\) for every column matrix X of size n x 1. Matrices B and C are square matrices of order n.

We can rearrange the equation \(BX = CX\) as follows:

\(BX - CX = 0\)

Using the distributive property of matrix multiplication \((B - C)X = BX - CX\), we can factor out the matrix X:

\((B - C)X = 0\)

Here, 0 represents the zero column matrix of size n x 1.

This equation \((B - C)X = 0\) must hold for *every* possible column matrix X of size n x 1.

Consider the standard basis vectors for X. These are column matrices where one element is 1 and all other elements are 0. For example, the first standard basis vector \(e_1\) is a column matrix with 1 in the first row and 0s elsewhere, \(e_2\) has 1 in the second row and 0s elsewhere, and so on, up to \(e_n\).

• If we take \(X = e_1\), the equation \((B - C)e_1 = 0\) means the first column of the matrix \((B - C)\) is the zero vector.
• If we take \(X = e_2\), the equation \((B - C)e_2 = 0\) means the second column of the matrix \((B - C)\) is the zero vector.
• Continuing this for all standard basis vectors \(e_i\) where \(i = 1, 2, \dots, n\), we find that every column of the matrix \((B - C)\) must be the zero vector.

A matrix where every column is the zero vector is the zero matrix. Therefore, \((B - C)\) must be the zero matrix.

\(B - C = 0\)

Adding C to both sides gives:

\(B = C\)

Thus, if \(BX = CX\) for every column matrix X, then \(B = C\). Statement 2 is correct.

Conclusion

Both Statement 1 and Statement 2 are correct properties concerning square matrices. Statement 1 demonstrates cancellation with a non-singular matrix, and Statement 2 shows that if two matrices produce the same result when multiplied by any vector, the matrices must be equal.

Based on the analysis, both statements are correct.