Square Matrix Properties

Scaling a determinant

Multiplying every entry of an $n \times n$ matrix by a constant $k$ scales the determinant by $k^n$, not just $k$:

If $A$ is a $3\times3$ matrix with $|A|=5$, then $|2A| = 2^3(5) = 40$.

Worked example: an idempotent matrix

If $A$ is a square matrix with $A^2 = A$ (called idempotent), simplify $(I-A)^3 + A$.

Solution: First find $(I-A)^2$:

So squaring $(I-A)$ gives back $(I-A)$ itself: $(I-A)$ is idempotent too. That means:

So $(I-A)^3 + A = (I-A) + A = I$.

Building a matrix from a formula

For a $2\times2$ matrix $A = [a_{ij}]$ where $a_{ij} = \dfrac{i}{j}$, find $A$.

Solution: Compute each entry directly from its row/column index:

A singular matrix and no solution

For a matrix equation $AX=B$ with square matrix $A$: if $|A|=0$ (A is singular) and $(\text{adj }A)B \ne 0$, the system has no solution at all: it's inconsistent. (If instead $(\text{adj }A)B = 0$, the system has infinitely many solutions.)