Questions: State whether each of the following statements is (always) true or (sometimes) false. If it is true, give a brief justification; if it is false, give a counterexample with an explanation. a) If A, B are 2 × 2 matrices with AB=0 and A ≠ 0, then B=0. b) If A, B are 2 × 2 matrices with AB=0 and A is invertible, then B=0.

State whether each of the following statements is (always) true or (sometimes) false. If it is true, give a brief justification; if it is false, give a counterexample with an explanation.
a) If A, B are 2 × 2 matrices with AB=0 and A ≠ 0, then B=0.
b) If A, B are 2 × 2 matrices with AB=0 and A is invertible, then B=0.

Transcript text: 5. * State whether each of the following statements is (always) true or (sometimes) false. If it is true, give a brief justification; if it is false, give a counterexample with an explanation. a) If $A, B$ are $2 \times 2$ matrices with $A B=0$ and $A \neq 0$, then $B=0$. b) If $A, B$ are $2 \times 2$ matrices with $A B=0$ and $A$ is invertible, then $B=0$.

Solution

Solution Steps

Solution Approach

a) This statement is sometimes false. A counterexample can be provided where $ A \neq 0 $ and $ B \neq 0 $ but $ AB = 0 $.

b) This statement is always true. If $ A $ is invertible, then multiplying both sides of $ AB = 0 $ by $ A^{-1} $ will show that $ B = 0 $.

Step 1: Analyze Statement (a)

The statement claims that if $ A $ and $ B $ are $ 2 \times 2 $ matrices such that $ AB = 0 $ and $ A \neq 0 $, then $ B $ must be $ 0 $.

To test this, we can consider the matrices: \[ A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \] Calculating $ AB $: \[ AB = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \] Since $ AB \neq 0 $, this does not serve as a valid counterexample. However, if we take: \[ B = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \] Then $ AB = 0 $ holds true, but $ B $ can also be non-zero while $ AB = 0 $. Thus, the statement is sometimes false.

Step 2: Analyze Statement (b)

The statement asserts that if $ A $ and $ B $ are $ 2 \times 2 $ matrices with $ AB = 0 $ and $ A $ is invertible, then $ B $ must be $ 0 $.

Since $ A $ is invertible, we can multiply both sides of the equation $ AB = 0 $ by $ A^{-1} $: \[ A^{-1}AB = A^{-1}0 \implies B = 0 \] This confirms that the statement is always true.