Q.Find the value of the following: For what values of $x : \begin{bmatrix} 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ x \end{bmatrix} = O$?

Idea: The product of a $1\times3$ row, a $3\times3$ matrix and a $3\times1$ column is a single number; set it to $0$.

Step 1 — row times matrix.

Step 2 — times the column. …

The left side is a $1\times3$ row times a $3\times3$ matrix times a $3\times1$ column. Checking shapes: $(1\times3)(3\times3)=(1\times3)$, then $(1\times3)(3\times1)=(1\times1)$ — a single number. The symbol $O$ here is that $1\times1$ zero, i.e. the scalar $0$. Multiply left-to-right.

Step 1 — row vector times the matrix

Each output entry is the row dotted with a column of the matrix:

• Column 1: $1\cdot1+2\cdot2+1\cdot1 = 1+4+1 = 6$
• Column 2: $1\cdot2+2\cdot0+1\cdot0 = 2$
• Column 3: $1\cdot0+2\cdot1+1\cdot2 = 0+2+2 = 4$

So the result is $\begin{bmatrix} 6 & 2 & 4 \end{bmatrix}$.

Step 2 — that row times the column …

Method: Evaluating a row $\times$ matrix $\times$ column product and solving

Use this for any expression of the form (row vector)(square matrix)(column vector) $= O$, where $O$ is a $1\times1$ zero, i.e. the scalar $0$.

Steps

Step 1: Confirm the shape collapses to $1\times 1$.

$(1\times n)(n\times n)(n\times 1) = 1\times 1$. So the whole product is a single number and $O$ means the scalar $0$ — one equation in the unknown.

Step 2: Multiply left-to-right (associativity lets you choose either grouping). …

Common Mistakes

Mistake 1: Reading $O$ as a matrix of some other size.

Why it's wrong: the product here is $1\times1$, so $O$ is the single scalar $0$, giving exactly one equation. Correct approach: track the shape — $(1\times3)(3\times3)(3\times1)=1\times1$.

Mistake 2: Multiplying entries position-by-position instead of row-by-column. …