Matrices and Determinants
Matrices and Determinants tests matrix algebra, determinant properties, invertibility, and solving linear systems. The exam expects correct order and conformability, not just arithmetic.
Core rules
Two matrices are equal only when same order and corresponding entries equal. Addition is entrywise; scalar multiplication multiplies every entry. AB is defined when columns of A equal rows of B, and generally AB≠BA. Transpose reverses order: (AB)ᵀ=BᵀAᵀ. A square matrix is singular if |A|=0. For a 2×2 matrix [[a,b],[c,d]], |A|=ad−bc. If |A|≠0, A⁻¹=(adj A)/|A|, and |adj A|=|A|ⁿ⁻¹ for order n. The system AX=B has a unique solution when |A|≠0.
Worked example
Let A=[[2,1],[1,3]] and solve 2x+y=7, x+3y=11. |A|=2·3−1·1=5. adj A=[[3,−1],[−1,2]], so A⁻¹=(1/5)[[3,−1],[−1,2]]. Then [x,y]ᵀ=A⁻¹[7,11]ᵀ=(1/5)[21−11, −7+22]ᵀ=(1/5)[10,15]ᵀ=[2,3]ᵀ. Thus x=2, y=3.
Traps
Never write (A+B)⁻¹=A⁻¹+B⁻¹; this is false. det(A+B) is not det A+det B. In a 3×3 expansion, watch the alternating signs. A zero determinant means no inverse, but the system may still be inconsistent or have infinitely many solutions. Check multiplication order when using inverse matrices.