Let and B = A10Then the sum of elements of the first column of B is

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Multiplication of Matrices

Question

Let $A = [\begin{array}{lll} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{array}]$ and B = A¹⁰
Then the sum of elements of the first column of B is

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$A = [\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{array}]$

A^{2} = [\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{array}] [\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{array}]

= [\begin{array}{ccc} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{array}]

Seeing the trend we can generalize that

A^{n} = [\begin{array}{ccc} 1 & 0 & 0 \\ n & 1 & 0 \\ \frac{(n + 1) n}{2} & n & 1 \end{array}]

A^{10} = [\begin{array}{ccc} 1 & 0 & 0 \\ 10 & 1 & 0 \\ 55 & 10 & 1 \end{array}]

Sum of the element of 1st column

1 + 10 + 55 = 66