Mathematical Logic and Matrices
Mathematical Logic and Matrices combines statement logic with matrix algebra. Logic questions test truth values, connectives, negation, converse, inverse, contrapositive, tautology and contradiction. Matrix questions test types of matrices, operations, determinants, inverses and solving linear systems.
Core rules
- A statement has a definite truth value. Connectives: p∧q is true only when both are true; p∨q is false only when both are false; p→q is false only when p is true and q is false; p↔q is true when truth values agree.
- Negation: ¬(p∧q)=¬p∨¬q and ¬(p∨q)=¬p∧¬q. The contrapositive of p→q is ¬q→¬p and is logically equivalent to it; the converse is q→p.
- Matrix addition requires equal order; multiplication AB requires columns of A = rows of B, and AB≠BA in general.
- For A=[[a,b],[c,d]], det A=ad-bc; if det A≠0, A^(-1)=(1/det A)[[d,-b],[-c,a]].
- A system AX=B has a unique solution X=A^(-1)B when det A≠0.
Worked example
Let A=[[2,1],[1,1]]. det A=2·1-1·1=1. Swap the diagonal entries and change the signs of the off-diagonal entries: A^(-1)=[[1,-1],[-1,2]]. Check: [[2,1],[1,1]][[1,-1],[-1,2]]=[[1,0],[0,1]].
Mark-losing traps
- Confusing converse with contrapositive.
- Treating or as exclusive when the exam uses inclusive or.
- Multiplying matrices in the wrong order.
- Forgetting to divide by the determinant in the inverse.
- Claiming an inverse exists when det A=0.