TOPICS

Unit-3(Theorems)

Matrices

**Theorem-1 :-** For any square matrix A with real number entries, A + A′ is a symmetric matrix and A – A′ is a skew symmetric matrix.

Proof : Let B = A + A′, then B′ = (A + A′)′ = A′ + (A′)′ [as (A + B)′ = A′ + B′] = A′ + A [as (A′)′ = A] = A + A′ [as A + B = B + A] = B Therefore B = A + A′ is a symmetric matrix Now let C = A – A′ C′ = (A – A′)′ = A′ – (A′)′ = A′ – A = – (A – A′) = – C Therefore C = A – A′ is a skew symmetric matrix.

**Theorem-2 :-** Any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.

Proof : Let A be a square matrix, then we can write A = 1/2(A + A') + 1/2(A - A') From the Theorem 1, we know that (A + A′) is a symmetric matrix and (A – A′) is a skew symmetric matrix. Since for any matrix A, (kA)′ = kA′, it follows that 1/2(A + A') is symmetric matrix and 1/2(A - A') is skew symmetric matrix. Thus, any square matrix can be expressed as the sum of a symmetric and a skew symmetric matrix.

**Theorem-3 :-** (Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique.

Proof : Let A = [aᵢⱼ] be a square matrix of order m. If possible, let B and C be two inverses of A. We shall show that B = C. Since B is the inverse of A AB = BA = I ... (1) Since C is also the inverse of A AC = CA = I ... (2) Thus B = BI = B (AC) = (BA) C = IC = C

**Theorem-4 :-** If A and B are invertible matrices of the same order, then (AB)^{-1} = B^{-1} A^{-1}.

Proof : From the definition of inverse of a matrix, we have (AB) (AB)^{-1}= 1 A^{-1}(AB) (AB)^{-1}= A^{-1}I [Pre multiplying both sides by A^{-1}] (A^{-1}A) B (AB)^{-1}= A^{-1}[Since A^{-1}I = A^{-1}] IB (AB)^{-1}= A^{-1}B (AB)^{-1}= A^{-1}B^{-1}B (AB)^{-1}= B^{-1}A^{-1}I (AB)^{-1}= B^{-1}A^{-1}Hence, (AB)^{-1}= B^{-1}A^{-1}

CLASSES