Question

What is the condition that is to be satisfied for the identity \((\mathrm{P}+\mathrm{Q})(\mathrm{P}-\mathrm{Q})=\mathrm{P}^{2}-\mathrm{Q}^{2}\) to be true for any two square matrices \(\mathrm{P}\) and \(\mathrm{Q}\) ? (1) The identity is always true. (2) \(\mathrm{PQ} \neq \mathrm{QP}\). (3) Both \(\mathrm{PQ}\) and \(\mathrm{QP}\) are not null matrices. (4) \(\mathrm{P}, \mathrm{Q}\) and \(\mathrm{PQ}\) are symmetric.

Short answer

(a) The identity is always true. (b) \(PQ \neq QP\) (c) Both \(PQ\) and \(QP\) are not null matrices. (d) \(P, Q\) and \(PQ\) are symmetric. Answer: None of the above options are correct. The correct condition is that \(PQ = QP\).

Step-by-step solution

Expand the given identity for matrices

Expand the identity \((P+Q)(P-Q)\) using matrix multiplication rules (distributive property): \((P+Q)(P-Q) = P(P-Q) + Q(P-Q) = PP - PQ + QP - QQ = P^2 - PQ + QP - Q^2\). Now we need to compare this expression with the given identity \(P^2 - Q^2\).

Analyze the middle terms

The expression \(P^2 - PQ + QP - Q^2\) has two middle terms, \(-PQ\) and \(+QP\), which are not present in the given identity \(P^2 - Q^2\). If the given identity is to be true, the middle terms should cancel each other out, meaning \(PQ = QP\). In other words, the two square matrices P and Q must commute. Now let's analyze the given options one by one.

Evaluate option (1)

Option (1) states that the identity is always true. However, this statement is false because the identity holds true only if \(PQ = QP\). Therefore, option (1) is incorrect.

Evaluate option (2)

Option (2) states that \(PQ \neq QP\). This statement is the opposite of the necessary condition \((PQ = QP)\) for the given identity to be true. Therefore, option (2) is incorrect.

Evaluate option (3)

Option (3) states that both \(PQ\) and \(QP\) are not null matrices. This implies that both \(PQ\) and \(QP\) should be non-zero. However, this statement does not guarantee that the given identity holds true. Therefore, option (3) is incorrect.

Evaluate option (4)

Option (4) states that \(P, Q\) and \(PQ\) are symmetric. This statement does not guarantee that the given identity holds true because it does not enforce the condition \(PQ = QP\). Therefore, option (4) is incorrect.

Conclusion

None of the given options are correct. The condition necessary for the identity \((P+Q)(P-Q) = P^2 - Q^2\) to be true for any two square matrices P and Q is that \(PQ = QP\).

Key concepts

Square Matrices

Square matrices are special types of matrices where the number of rows is equal to the number of columns. This symmetrical structure has unique properties and applications in mathematics and programming. For a matrix to be square, if it has 'n' rows, it must also have 'n' columns, making it an 'n x n' matrix.
Square matrices are essential in linear algebra because they allow for operations such as calculating determinants, finding inverses, and defining the notion of eigenvalues and eigenvectors.

• The diagonal elements of a square matrix (from top left to bottom right) play a critical role in its properties.
• Operations specifically applicable to square matrices include trace calculation, where the sum of diagonal elements is taken.

The significance of square matrices in various mathematical operations cannot be understated, particularly in solving systems of linear equations and representing linear transformations.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra where two matrices are multiplied to produce a third matrix. Unlike regular multiplication, this process involves a row-by-column rule:

• To multiply matrix A by matrix B, the number of columns in A must equal the number of rows in B.
• The resulting matrix will have dimensions equal to the number of rows of A and the number of columns of B.
• This method involves calculating each element in the resulting matrix by taking the dot product of the rows of the first matrix with the columns of the second.

When dealing with square matrices, multiplication can reveal whether matrices commute, i.e., if \(PQ = QP\).
If two matrices commute, their product is independent of the order; however, not all matrices commute, highlighting their complex interplay during operations.

Identity in Matrices

The identity matrix, denoted as \(I_n\), acts as the multiplicative identity in matrix algebra, much like the number 1 in regular arithmetic operations. This matrix is crucial for understanding matrix commutation and transformations.

• An identity matrix is a square matrix with 1s on the main diagonal and 0s elsewhere.
• For any matrix \(A\) of compatible dimensions, the equation \(AI = IA = A\) holds true.
• In problems involving matrix multiplication, the identity matrix helps ascertain if transformations have been correctly applied without altering the original matrix.

The role of an identity matrix simplifies processes such as finding inverses and confirming orthogonality, making it highly relevant in both theoretical and applied scenarios. Understanding the identity in matrices provides clarity, especially when verifying the results of complex matrix equations.