Understanding Matrix Properties: Commutativity and Null Matrix Product

This question asks us to analyze two properties related to matrix multiplication for the given matrices A and B. We need to determine if matrix A and matrix B commute (i.e., if AB = BA) and if their product AB results in a null matrix.

The given matrices are:

\({\rm{A}} = \left[ {\begin{array}{*{20}{c}} 1&1&{ - 1}\\ 2&{ - 3}&4\\ 3&{ - 2}&3 \end{array}} \right]\)

\({\rm{B}} = \left[ {\begin{array}{*{20}{c}} { - 1}&{ - 2}&{ - 1}\\ 6&{12}&6\\ 5&{10}&5 \end{array}} \right]\)

Both A and B are 3x3 matrices, so matrix multiplication AB and BA is possible, and the resulting matrices will also be 3x3.

Calculating the Matrix Product AB

To check statement 2 first, let's compute the product AB. The entry in the i-th row and j-th column of AB is obtained by multiplying the i-th row of A by the j-th column of B element-wise and summing the results.

Let \({\rm{AB}} = [C_{ij}]\).

• \(C_{11} = (1)(-1) + (1)(6) + (-1)(5) = -1 + 6 - 5 = 0\)
• \(C_{12} = (1)(-2) + (1)(12) + (-1)(10) = -2 + 12 - 10 = 0\)
• \(C_{13} = (1)(-1) + (1)(6) + (-1)(5) = -1 + 6 - 5 = 0\)

First row of AB is \(\left[ {\begin{array}{*{20}{c}} 0&0&0 \end{array}} \right]\).

• \(C_{21} = (2)(-1) + (-3)(6) + (4)(5) = -2 - 18 + 20 = 0\)
• \(C_{22} = (2)(-2) + (-3)(12) + (4)(10) = -4 - 36 + 40 = 0\)
• \(C_{23} = (2)(-1) + (-3)(6) + (4)(5) = -2 - 18 + 20 = 0\)

Second row of AB is \(\left[ {\begin{array}{*{20}{c}} 0&0&0 \end{array}} \right]\).

• \(C_{31} = (3)(-1) + (-2)(6) + (3)(5) = -3 - 12 + 15 = 0\)
• \(C_{32} = (3)(-2) + (-2)(12) + (3)(10) = -6 - 24 + 30 = 0\)
• \(C_{33} = (3)(-1) + (-2)(6) + (3)(5) = -3 - 12 + 15 = 0\)

Third row of AB is \(\left[ {\begin{array}{*{20}{c}} 0&0&0 \end{array}} \right]\).

Thus, the matrix product AB is:

\({\rm{AB}} = \left[ {\begin{array}{*{20}{c}} 0&0&0\\ 0&0&0\\ 0&0&0 \end{array}} \right]\)

This is indeed a null matrix (a matrix where all entries are zero). Therefore, statement 2, "AB is a null matrix," is correct.

Checking for Commutativity (A and B commute?)

For statement 1, A and B commute if and only if AB = BA. We have already calculated AB. Now, let's compute BA.

Let \({\rm{BA}} = [D_{ij}]\).

• \(D_{11} = (-1)(1) + (-2)(2) + (-1)(3) = -1 - 4 - 3 = -8\)
• \(D_{12} = (-1)(1) + (-2)(-3) + (-1)(-2) = -1 + 6 + 2 = 7\)
• \(D_{13} = (-1)(-1) + (-2)(4) + (-1)(3) = 1 - 8 - 3 = -10\)

The first row of BA is \(\left[ {\begin{array}{*{20}{c}} { - 8}&7&{ - 10} \end{array}} \right]\).

Since the first entry \(D_{11}\) of BA is -8, and the corresponding entry \(C_{11}\) of AB is 0, we can immediately see that AB is not equal to BA. \(0 \neq -8\).

Therefore, statement 1, "A and B commute," is incorrect.

Conclusion on Statements

Based on our calculations:

• Statement 1: A and B commute (AB = BA) - False
• Statement 2: AB is a null matrix - True

The correct option should indicate that only statement 2 is correct.

Reviewing the Options

Let's look at the provided options:

1. 1 only (Incorrect, statement 1 is false)
2. 2 only (Correct, statement 2 is true and statement 1 is false)
3. Both 1 and 2 (Incorrect, statement 1 is false)
4. Neither 1 nor 2 (Incorrect, statement 2 is true)

The option that correctly reflects our findings is "2 only".

Matrix Commutativity and Null Products

In general, matrix multiplication is not commutative, meaning for two matrices A and B, AB is usually not equal to BA. Matrices A and B commute if and only if AB = BA.

A null matrix (or zero matrix) is a matrix where every entry is zero. It is denoted by 0 (often with subscripts indicating its dimensions, e.g., \(0_{3 \times 3}\)). When the product of two matrices is a null matrix, it doesn't necessarily mean that either of the original matrices is a null matrix. This is a key difference from multiplication of real numbers, where if \(ab=0\), then either \(a=0\) or \(b=0\) (or both). In matrix algebra, you can have AB = 0 even if A \(\neq\) 0 and B \(\neq\) 0, as demonstrated in this problem.

Revision Table: Key Matrix Concepts

Additional Information on Matrix Multiplication

Matrix multiplication requires the number of columns in the first matrix to be equal to the number of rows in the second matrix. For matrices A (m x n) and B (p x q), the product AB is defined only if n = p, and the resulting matrix AB has dimensions m x q. The product BA is defined only if q = m, and the resulting matrix BA has dimensions p x n.

In this specific problem, both A and B are 3x3 matrices. So, the number of columns in A (3) equals the number of rows in B (3), making AB defined and a 3x3 matrix. Similarly, the number of columns in B (3) equals the number of rows in A (3), making BA defined and a 3x3 matrix.

The fact that AB is a null matrix while neither A nor B is a null matrix is an important property of matrix algebra, highlighting differences from scalar arithmetic.