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AM,GM,HM Inequality

Let A=[aij] b...

Question

Let $A = [a_{i j}]$ be a real matrix of order $3 \times 3$ , such that $a_{i 1} + a_{i 2} + a_{i 3} = 1$ , for $i = 1, 2, 3.$ Then, the sum of all the entries of the matrix $A^{3}$ is equal to:

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Solution

Let a matrix $B = ⎡ ⎢ ⎣ \begin{matrix} 111 \end{matrix} ⎤ ⎥ ⎦$
$∴ A \cdot B = B$
Sum of all entries of $A^{3}$ is equal to the only element of $B^{T} . A^{3} . B$
$∴ B^{T} . A^{3} . B = B^{T} . A^{2} . (A B) = B^{T} . A^{2} . B = B^{T} . B = [3]_{1 \times 1}$

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A = [a_{i j}]

be a square matrix of order

3

a_{i j} = 2^{j - i}

i, j = 1, 2, 3

. Then, the matrix

A^{2} + A^{3} + \dots + A^{10}

A

be a square matrix of order

3

whose all entries are

1

I_{3}

be the identity matrix of order

3

. Then the matrix

A - 3 I_{3}

A = {[a_{i j}]}_{m \times n}

be a matrix such that

a_{i j} = 1

for all i,j. Then,

A

3 \times 3

matrix such that

a_{11} = a_{33} = 2

and all the other

a_{i j} = 1

A^{- 1} = x A^{2} + y A + z I

, then the value of

(x + y + z)

I

is a unit matrix of order

3

A = [a_{i j}]

is scalar matrix of order

n \times n

a_{i j} = k

i

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