Question

If M and N are two 3x3 diagonal matrices, then which of the following is/are true?

• A. MN is diagonal matrix
• B. MN = NM
• C. M² + N² is a diagonal matrix
• D. MN+NM=0

Answer 1

Answer: (A), (B), (C)

A, B, C

Answer 2

Let M and N be two 3x3 diagonal matrices. This means they can be represented as:

$M = \begin{pmatrix} m_{11} & 0 & 0 \\ 0 & m_{22} & 0 \\ 0 & 0 & m_{33} \end{pmatrix}$

$N = \begin{pmatrix} n_{11} & 0 & 0 \\ 0 & n_{22} & 0 \\ 0 & 0 & n_{33} \end{pmatrix}$

Let's evaluate each option:

(A) MN is a diagonal matrix

Let's compute the product MN:

$MN = \begin{pmatrix} m_{11} & 0 & 0 \\ 0 & m_{22} & 0 \\ 0 & 0 & m_{33} \end{pmatrix} \begin{pmatrix} n_{11} & 0 & 0 \\ 0 & n_{22} & 0 \\ 0 & 0 & n_{33} \end{pmatrix}$

$MN = \begin{pmatrix} m_{11}n_{11} & 0 & 0 \\ 0 & m_{22}n_{22} & 0 \\ 0 & 0 & m_{33}n_{33} \end{pmatrix}$

The resulting matrix has non-zero elements only on its main diagonal. Therefore, MN is a diagonal matrix. Statement (A) is true.

(B) MN = NM

From (A), we have $MN = \begin{pmatrix} m_{11}n_{11} & 0 & 0 \\ 0 & m_{22}n_{22} & 0 \\ 0 & 0 & m_{33}n_{33} \end{pmatrix}$.

Now, let's compute NM:

$NM = \begin{pmatrix} n_{11} & 0 & 0 \\ 0 & n_{22} & 0 \\ 0 & 0 & n_{33} \end{pmatrix} \begin{pmatrix} m_{11} & 0 & 0 \\ 0 & m_{22} & 0 \\ 0 & 0 & m_{33} \end{pmatrix}$

$NM = \begin{pmatrix} n_{11}m_{11} & 0 & 0 \\ 0 & n_{22}m_{22} & 0 \\ 0 & 0 & n_{33}m_{33} \end{pmatrix}$

Since the multiplication of real numbers is commutative ($m_{ii}n_{ii} = n_{ii}m_{ii}$), we can see that $MN = NM$. Statement (B) is true.

(C) M² + N² is a diagonal matrix

First, let's find M²:

$M^2 = M \cdot M = \begin{pmatrix} m_{11} & 0 & 0 \\ 0 & m_{22} & 0 \\ 0 & 0 & m_{33} \end{pmatrix} \begin{pmatrix} m_{11} & 0 & 0 \\ 0 & m_{22} & 0 \\ 0 & 0 & m_{33} \end{pmatrix} = \begin{pmatrix} m_{11}^2 & 0 & 0 \\ 0 & m_{22}^2 & 0 \\ 0 & 0 & m_{33}^2 \end{pmatrix}$

M² is a diagonal matrix.

Similarly, N² will be:

$N^2 = N \cdot N = \begin{pmatrix} n_{11}^2 & 0 & 0 \\ 0 & n_{22}^2 & 0 \\ 0 & 0 & n_{33}^2 \end{pmatrix}$

N² is a diagonal matrix.

Now, let's compute M² + N²:

$M^2 + N^2 = \begin{pmatrix} m_{11}^2 & 0 & 0 \\ 0 & m_{22}^2 & 0 \\ 0 & 0 & m_{33}^2 \end{pmatrix} + \begin{pmatrix} n_{11}^2 & 0 & 0 \\ 0 & n_{22}^2 & 0 \\ 0 & 0 & n_{33}^2 \end{pmatrix} = \begin{pmatrix} m_{11}^2 + n_{11}^2 & 0 & 0 \\ 0 & m_{22}^2 + n_{22}^2 & 0 \\ 0 & 0 & m_{33}^2 + n_{33}^2 \end{pmatrix}$

The resulting matrix has non-zero elements only on its main diagonal. Therefore, M² + N² is a diagonal matrix. Statement (C) is true.

(D) MN + NM = 0

From (B), we know that MN = NM.

So, MN + NM = MN + MN = 2MN.

We found in (A) that $MN = \begin{pmatrix} m_{11}n_{11} & 0 & 0 \\ 0 & m_{22}n_{22} & 0 \\ 0 & 0 & m_{33}n_{33} \end{pmatrix}$.

For 2MN to be the zero matrix (0), all its elements must be zero. This means $2m_{ii}n_{ii} = 0$ for i=1,2,3. This implies $m_{ii}n_{ii} = 0$ for all i. This is not generally true for any two diagonal matrices M and N (e.g., if M and N are identity matrices, MN is not zero).

Therefore, MN + NM = 0 is not generally true. Statement (D) is false.

The true statements are (A), (B), and (C).

Explanation:

1. Diagonal Matrix Definition: A diagonal matrix has all non-diagonal elements equal to zero.
2. Product of Diagonal Matrices (MN): When two diagonal matrices are multiplied, the resulting matrix is also a diagonal matrix. The diagonal elements of the product are the products of the corresponding diagonal elements of the original matrices. $(MN)_{ii} = m_{ii}n_{ii}$, and $(MN)_{ij} = 0$ for $i \neq j$. This proves (A).
3. Commutativity of Diagonal Matrices (MN = NM): Since the diagonal elements are scalar products ($m_{ii}n_{ii} = n_{ii}m_{ii}$), and all off-diagonal elements are zero for both MN and NM, it follows that MN = NM. This proves (B).
4. Square of a Diagonal Matrix (M², N²): The square of a diagonal matrix is also a diagonal matrix, with diagonal elements being the squares of the original diagonal elements. $(M^2)_{ii} = m_{ii}^2$, and $(M^2)_{ij} = 0$ for $i \neq j$. Similarly for N².
5. Sum of Diagonal Matrices (M² + N²): The sum of two diagonal matrices is a diagonal matrix. The diagonal elements of the sum are the sums of the corresponding diagonal elements of the matrices being added. $(M^2 + N^2)_{ii} = m_{ii}^2 + n_{ii}^2$, and $(M^2 + N^2)_{ij} = 0$ for $i \neq j$. This proves (C).
6. MN + NM = 0: Since MN = NM, this expression simplifies to 2MN. For 2MN to be the zero matrix, MN must be the zero matrix. This is not generally true for arbitrary diagonal matrices M and N (e.g., if M and N are non-zero identity matrices). This disproves (D).