Let A = . If M and N are two matrices given by M = and N = then MN2 is :

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Multiplication of Matrices

Question

Let A = $[\begin{matrix} 0 & - 2 \\ 2 & 0 \end{matrix}]$ . If M and N are two matrices given by M = $\sum_{k = 1}^{10} A^{2 k}$ and N = $\sum_{k = 1}^{10} A^{2 k - 1}$ then MN² is :

a non-identify symmetric matrix

an identify matrix

a skew-symmetric matrix

neither symmetric nor skew-symmetric matrix

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A² = $[\begin{matrix} 0 & - 2 \\ 2 & 0 \end{matrix}] . [\begin{matrix} 0 & - 2 \\ 2 & 0 \end{matrix}] = [\begin{matrix} - 4 & 0 \\ 0 & - 4 \end{matrix}] = 4 Ι .$ (symmetric)

& A³ = – 4A (skew symmetric)

⇒ M = $\sum_{k = 1}^{10} A^{2 k}$ = [(– 4) + (– 4)² + (– 4)³ + …. + (– 4)¹⁰] I

= – 4λ I is symmetric

⇒ N = $\sum_{k = 1}^{10} A^{2 k - 1}$ = A [1 + (– 4) + (– 4)³ + …. + (– 4)⁹] I

= λ A is skew symmetric

Where λ = {1 + (– 4) + (– 4)³ + …. + (– 4)⁹}

Now MN = – 4λ² A = NM

⇒ MN² = (MN)N = (NM)N = N(MN) =N(NM) = N²M

Hence (MN²)^T = (N²)^T M^T = (N^T)² M^T = (– N)² M = N²M

⇒ MN² is symmetric matrix