Exam Questions - Matrix proofs

Matrix Proofs Revision Content

Basics of Matrix Proofs

A matrix is a rectangular array of numbers, symbols, or expressions.
When solving problems, you often need to prove certain properties about matrices - these are matrix proofs.
Key techniques for these proofs include using definitions and properties of matrix operations, and specifically using properties of addition, subtraction, multiplication, and scalar multiplication.

Matrix Equality

Matrix equality is important: two matrices are equal if they have the same size and their corresponding elements are equal. This definition is often used in matrix proofs.

Matrix Operations

Remember the properties of matrix addition and subtraction: they’re commutative (A + B = B + A) and associative ([A + B] + C = A + [B + C]). Also, there exists additive identity (A + 0 = A) and additive inverse property (A + -A = 0).
Matrix multiplication, however, isn’t commutative (AB ≠ BA). It’s associative and distributes over addition. Do not forget the existence of a multiplicative identity (AI = IA = A).
For scalar multiplication, remember that constants can go anywhere in the product (kA = Ak) and it distributes over addition of matrices and scalars.

Special Matrices

Special matrices such as the identity matrix (remains the same when multiplied with any other matrix), the zero matrix (all elements are zero), and diagonal matrices (only diagonal elements are non-zero) often play key roles in proofs.
For transpose of a matrix (A^T), (A^T)^T = A and (A + B)^T = A^T + B^T. For product of matrices, (AB)^T = B^T A^T.

Inverse Matrices

An inverse matrix is such that when it is multiplied with the original matrix, the result is the identity matrix.
A key property to remember is that (AB)^-1 = B^-1 A^-1.

Determinants and Adjoint

The determinant of a matrix is a special number that can be calculated from a square matrix.
The adjoint of a matrix is the transpose of the cofactor matrix.
Both determinant and adjoint have unique properties used extensively in matrix proofs.

This outline covers the key concepts needed for matrix proofs. Practice these principles until you’re comfortable using them to solve problems and perform proofs. Regular review and exercise will solidify your understanding and enable you to tackle more difficult problems.