Matrix operations — when can you add, multiply, find inverse? Conditions flowchart

Question

Given two matrices $A$ and $B$ , when can we add them, multiply them, and find their inverses? State all necessary conditions with examples.

(CBSE 12 + JEE Main — theory + MCQ)

Solution — Step by Step

$A + B$ is defined ONLY when $A$ and $B$ have the same order (same number of rows AND columns).

If $A$ is $2 \times 3$ and $B$ is $2 \times 3$ : addition is possible, result is $2 \times 3$ .

If $A$ is $2 \times 3$ and $B$ is $3 \times 2$ : addition is NOT possible.

$AB$ is defined when the number of columns of $A$ equals the number of rows of $B$ .

If $A$ is $m \times n$ and $B$ is $n \times p$ : multiplication is possible, result is $m \times p$ .

If $A$ is $2 \times 3$ and $B$ is $3 \times 4$ : $AB$ exists and is $2 \times 4$ .

Note: $AB$ may exist even when $BA$ does not. And even if both exist, generally $AB \neq BA$ .

$A^{-1}$ exists ONLY when:

$A$ is a square matrix ( $n \times n$ )
$|A| \neq 0$ (determinant is non-zero, i.e., $A$ is non-singular)

If $|A| = 0$ , the matrix is singular and has no inverse.

Formula: $A^{-1} = \frac{1}{|A|} \cdot \text{adj}(A)$

Let $A = \begin{pmatrix} 2 & 1 \\ 4 & 3 \end{pmatrix}$

$|A| = 2(3) - 1(4) = 6 - 4 = 2 \neq 0$ , so inverse exists.

$\text{adj}(A) = \begin{pmatrix} 3 & -1 \\ -4 & 2 \end{pmatrix}$

$A^{-1} = \frac{1}{2}\begin{pmatrix} 3 & -1 \\ -4 & 2 \end{pmatrix} = \begin{pmatrix} 3/2 & -1/2 \\ -2 & 1 \end{pmatrix}$

flowchart TD
    A["Matrix Operation Required"] --> B{"Which operation?"}
    B -- Addition --> C{"Same order m×n?"}
    C -- Yes --> D["Add element-wise. Result: m×n"]
    C -- No --> E["NOT POSSIBLE"]
    B -- Multiplication AB --> F{"Cols of A = Rows of B?"}
    F -- Yes --> G["Multiply. Result: rows_A × cols_B"]
    F -- No --> E
    B -- Inverse --> H{"Is A square?"}
    H -- No --> E
    H -- Yes --> I{"Is det A ≠ 0?"}
    I -- Yes --> J["Inverse exists: adj(A)/det(A)"]
    I -- No --> K["Singular — no inverse"]

Why This Works

Matrix addition requires same dimensions because we add corresponding elements — if the matrices have different sizes, there is no “corresponding element” for every position.

Matrix multiplication requires columns of $A$ = rows of $B$ because each entry of $AB$ is computed as a dot product of a row of $A$ with a column of $B$ . These vectors must have the same length for the dot product to work.

The inverse exists only for square, non-singular matrices because inverting means “undoing” the transformation. A singular matrix squishes space into a lower dimension (determinant = 0), losing information — there is no way to undo that.

Alternative Method

For $2 \times 2$ matrices, there is a shortcut for the inverse. For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ : swap $a$ and $d$ , negate $b$ and $c$ , divide by $(ad - bc)$ . Result: $A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ . This is faster than computing adj $(A)$ separately.

Common Mistake

Students assume that if $AB$ exists, then $BA$ also exists. This is false. If $A$ is $2 \times 3$ and $B$ is $3 \times 5$ , then $AB$ is $2 \times 5$ but $BA$ requires 5 columns in $B$ to match 2 rows of $A$ — which fails since $B$ has 3 rows, not 2. Always check dimensions before multiplying.