Exercise · ZW-MATRIX-MULTIPLICATION-010 · ★★★ Advanced
Which pair of matrices \(A,B\), both nonzero, satisfies \(AB=0\)?
Choose one option.
Show answer
- C. \(A=\begin{pmatrix}1&0\\0&0\end{pmatrix},\ B=\begin{pmatrix}0&0\\0&1\end{pmatrix}\)
Explanation
With \(A=\operatorname{diag}(1,0)\) and \(B=\operatorname{diag}(0,1)\), \(AB=0\). Unlike numbers, a zero product does not force a zero factor.
Review the concept: Matrix multiplication
The product \(AB\) of two matrices \(A\) and \(B\), when \(A\) has as many columns as \(B\) has rows, is the matrix whose \((i,j)\) entry combines row \(i\) of \(A\) with column \(j\) of \(B\).
\[(AB)_{ij}=\sum_{k=1}^{n}a_{ik}b_{kj}\]
More practice problems: matrix multiplication
- ZW-MATRIX-MULTIPLICATION-001 If A is 2× 3 and B is 3× 4, what is the size of AB?
- ZW-MATRIX-MULTIPLICATION-002 Compute AB for A=[1 2; 3 4] and B=[0 1; 1 0].
- ZW-MATRIX-MULTIPLICATION-003 Let A be 3× 4 and B be 4× 2. Which product is defined?
- ZW-MATRIX-MULTIPLICATION-004 For A=[2 -1; 0 3] and B=[1 4; 5 2], what is the entry (AB)_21?
- ZW-MATRIX-MULTIPLICATION-005 For A=[1 1; 0 1] and B=[1 0; 1 1], compute AB-BA.
- ZW-MATRIX-MULTIPLICATION-006 If I is the 2× 2 identity and A=[3 -2; 1 5], what is AI?
- ZW-MATRIX-MULTIPLICATION-007 Compute Amathbf v for A=[1 2 0; 0 1 -1] and mathbf v=[2; 1; 3].
- ZW-MATRIX-MULTIPLICATION-008 Which of these equalities holds for matrices A,B,C whose products are defined?