Q.Find $AB$, if $A = \begin{bmatrix} 0 & -1 \\ 0 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 5 \\ 0 & 0 \end{bmatrix}$.

Concept: Matrix Multiplication Compatibility — For $AB$ to be defined, the number of columns in $A$ must equal the number of rows in $B$. Here $A$ is $2 \times 2$ and $B$ is $2 \times 2$, so $AB$ is defined and will be $2 \times 2$.

Step 1: Multiply the first row of $A$ by each column of $B$:

• $(0)(3) + (-1)(0) = 0$
• $(0)(5) + (-1)(0) = 0$ …

Why this works: Matrix Multiplication Compatibility

Matrix multiplication isn't just about multiplying numbers — it's about combining the rows of the first matrix with the columns of the second. For $AB$ to be defined, the number of columns in $A$ must match the number of rows in $B$. Here:

• $A$ is $2 \times 2$ (2 rows, 2 columns)
• $B$ is $2 \times 2$ (2 rows, 2 columns)

The inner dimensions (2 and 2) match, so the product exists. The outer dimensions (2 and 2) tell us the result is also $2 \times 2$.

Step-by-step calculation

1. Entry $(1,1)$ — first row of $A$, first column of $B$:

• Row 1 of $A$: $[0, -1]$
• Column 1 of $B$: $\begin{bmatrix} 3 \\ 0 \end{bmatrix}$
• Dot product: $(0)(3) + (-1)(0) = 0 + 0 = 0$

2. Entry $(1,2)$ — first row of $A$, second column of $B$:

• Row 1 of $A$: $[0, -1]$
• Column 2 of $B$: $\begin{bmatrix} 5 \\ 0 \end{bmatrix}$
• Dot product: $(0)(5) + (-1)(0) = 0 + 0 = 0$

3. Entry $(2,1)$ — second row of $A$, first column of $B$:

• Row 2 of $A$: $[0, 2]$
• Column 1 of $B$: $\begin{bmatrix} 3 \\ 0 \end{bmatrix}$ …

Method: Multiplying matrices where a product turns out to be zero

Use this for any $AB$; it also shows how a nonzero $A$ and nonzero $B$ can multiply to the zero matrix.

Steps

Step 1: Confirm compatibility and result order.

Columns of $A$ must equal rows of $B$; the product's order is rows of $A$ by columns of $B$.

Step 2: Compute each entry as a dot product. …

Common Mistakes

Mistake 1: Assuming $AB = O$ forces $A = O$ or $B = O$.

Why it's wrong: unlike numbers, two nonzero matrices can multiply to the zero matrix (they are "zero divisors"). Correct approach: compute the product; a zero result is valid.

Mistake 2: Multiplying entry-by-entry instead of row-by-column.

Why it's wrong: that is not matrix multiplication. Correct approach: use dot products of rows and columns. …