Matrices and Determinants
Matrices and Determinants tests algebra of matrices, transpose and special matrices, evaluation and properties of determinants, adjoint and inverse, and solution of linear systems.
Core rules
- Matrix addition requires the same order; multiplication AB requires columns of A equal rows of B, and AB is usually not BA.
- For A=[[a,b],[c,d]], det A=ad-bc. A is singular exactly when det A=0.
- det(AB)=det A det B; interchanging two rows changes the sign; a row of zeros gives determinant zero.
- A inverse=(1/det A) adj A, valid only when det A is nonzero. adj A has cofactors with transposed positions and alternating signs.
Worked example
Let A=[[2,1],[1,3]]. Then det A=2×3-1×1=5. The cofactor matrix is [[3,-1],[-1,2]], so adj A=[[3,-1],[-1,2]]. Hence A inverse=(1/5)[[3,-1],[-1,2]]. Check: A times this matrix gives the identity.
Traps
- Never assume (A+B) inverse=A inverse+B inverse; inverses do not add.
- det(A+B) is not det A+det B in general.
- Watch cofactor signs: the (1,2) and (2,1) entries receive minus signs.
- A matrix can have a zero determinant and still have nonzero entries; it is then not invertible.
- In solving AX=B, multiply by A inverse on the left, not the right.