Properties of matrices — A/L Combined Mathematics

What must I remember about Properties of matrices?

• Definition of a Matrix: A rectangular array of real numbers arranged in m rows and n columns, having order (dimension) m × n.
• Matrix Equality: Two matrices A = [a_ij] and B = [b_ij] are equal if and only if they have the SAME order and corresponding entries are identical: a_ij = b_ij for all i, j.
• Matrix Addition and Subtraction: Defined only for matrices of identical order m × n: [a_ij] ± [b_ij] = [a_ij ± b_ij]; satisfies commutative and associative laws.
• Scalar Multiplication: Multiplication of matrix A by scalar k multiplies every entry by k: k*A = [k * a_ij]; distributive over matrix addition: k(A + B) = kA + kB.
• Zero (Null) Matrix O: Matrix of any dimension whose entries are all zero, serving as the additive identity: A + O = A.

Test yourself (1 minute)

1. Matrix multiplication compatibility condition

   The product AB is defined if and only if number of columns in A equals number of rows in B; if A is m × p and B is p × n, then AB is m × n.

2. Non-commutativity of matrix multiplication

   In general, AB ≠ BA; matrix multiplication is associative (A(BC) = (AB)C) and distributive (A(B + C) = AB + AC), but strictly NON-COMMUTATIVE.

3. Zero divisor property

   AB = O does NOT imply A = O or B = O; non-zero matrices can multiply to produce the zero matrix.

Practise Properties of matrices questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 24

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Year 2 · Term 1