Problem 23 - Algebra for JEE ADVANCED

Question

If \(D_{1}\) and \(D_{2}\) are two \(3 \times 3\) diagonal matrices, then which of the following is/are true? a. \(D_{1} D_{2}\) is diagonal matrix b. \(D_{1} D_{2}=D_{2} D_{1}\) c. \(D_{1}^{2}+D_{2}^{2}\) is a diagonal matrix d. none of these

Step-by-Step Solution

Answer

a, b, and c are true.

Step 1: Understanding Diagonal Matrices

A diagonal matrix is a square matrix in which all elements outside the main diagonal are zero (i.e., only the elements \(a_{ii}\) are non-zero). For two such matrices, \(D_1\) and \(D_2\), each having diagonal elements, all other entries are zero.

Step 2: Multiply Two Diagonal Matrices

When multiplying two diagonal matrices \(D_1 = \begin{bmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{bmatrix}\) and \(D_2 = \begin{bmatrix} d & 0 & 0 \ 0 & e & 0 \ 0 & 0 & f \end{bmatrix}\), the product \(D_1D_2\) results in another diagonal matrix \(\begin{bmatrix} ad & 0 & 0 \ 0 & be & 0 \ 0 & 0 & cf \end{bmatrix}\), since each element \(a_{ij} imes d_{ij} = 0\) except along the diagonal.

Step 3: Check Commutativity of Diagonal Matrices

Matrix multiplication of diagonal matrices is commutative, meaning \(D_1D_2 = D_2D_1\). This occurs because both products will have the form \(\begin{bmatrix} ad & 0 & 0 \ 0 & be & 0 \ 0 & 0 & cf \end{bmatrix}\). The result will be identical, as the order of multiplication of respective diagonal elements does not affect the product.

Step 4: Check Sums of Squared Matrices

For \(D_1^2 = \begin{bmatrix} a^2 & 0 & 0 \ 0 & b^2 & 0 \ 0 & 0 & c^2 \end{bmatrix}\) and \(D_2^2 = \begin{bmatrix} d^2 & 0 & 0 \ 0 & e^2 & 0 \ 0 & 0 & f^2 \end{bmatrix}\), the sum \(D_1^2 + D_2^2\) results in \(\begin{bmatrix} a^2 + d^2 & 0 & 0 \ 0 & b^2 + e^2 & 0 \ 0 & 0 & c^2 + f^2 \end{bmatrix}\), which is also a diagonal matrix.

Key Concepts

Commutativity of Matrices

When we discuss the commutativity of matrices, we refer to whether the multiplication of two matrices results in the same product regardless of the order in which they are multiplied. In general, matrix multiplication is not commutative, meaning that for most matrices \(A\) and \(B\), \(AB eq BA\). However, there are special cases where commutativity does occur.

One such special case is when dealing with diagonal matrices. A diagonal matrix is a square matrix in which only the diagonal elements are non-zero, while all off-diagonal elements are zero. For two diagonal matrices \(D_1\) and \(D_2\), the product \(D_1D_2\) will equal \(D_2D_1\).

This is because the multiplication of diagonal matrices only involves multiplying their respective diagonal entries. Thus, for diagonal matrices, the property \(D_1D_2 = D_2D_1\) holds, demonstrating commutativity in this context.

Properties of Diagonal Matrices

Diagonal matrices have unique properties that often simplify matrix operations. One key property is their simple multiplication rule: when you multiply two diagonal matrices, the product is another diagonal matrix. This is due to the nature of their elements—only diagonal elements need to be multiplied together, keeping the off-diagonal elements zero.

• Multiplication: For diagonal matrices \(D_1 = \begin{bmatrix} a & 0 & 0\ 0 & b & 0\ 0 & 0 & c\end{bmatrix}\) and \(D_2 = \begin{bmatrix} d & 0 & 0\ 0 & e & 0\ 0 & 0 & f\end{bmatrix}\), the product \(D_1D_2\) is \(\begin{bmatrix} ad & 0 & 0\ 0 & be & 0\ 0 & 0 & cf\end{bmatrix}\).
• Identity: The identity matrix is a special kind of diagonal matrix where all diagonal elements are one. Multiplying any diagonal matrix by the identity matrix leaves the original matrix unchanged.

These properties make calculations more efficient when working with diagonal matrices.

Matrix Addition

Adding matrices involves adding corresponding elements from each matrix. For diagonal matrices, the addition is straightforward since most elements are zero.

Consider two diagonal matrices \(D_1 = \begin{bmatrix} a & 0 & 0\ 0 & b & 0\ 0 & 0 & c\end{bmatrix}\) and \(D_2 = \begin{bmatrix} d & 0 & 0\ 0 & e & 0\ 0 & 0 & f\end{bmatrix}\). Their sum, \(D_1 + D_2\), results in \(\begin{bmatrix} a + d & 0 & 0\ 0 & b + e & 0\ 0 & 0 & c + f\end{bmatrix}\).

• Preservation of Form: The sum of two diagonal matrices is another diagonal matrix.
• Simplicity: Since only the diagonal elements are involved in the addition, the computation is typically simpler and faster than for non-diagonal matrices.

This property is particularly useful when dealing with operations involving multiple matrices, as it reduces complexity.