Solving for the Sum of Entries in Matrix $A^3$
We are given a $3\times3$ real matrix $A = [a_{ij}]$ with the property that the sum of the entries in each row is 1. Mathematically, this means:
$ a_{i1} + a_{i2} + a_{i3} = 1 \quad \text{for } i=1, 2, 3 $
We need to find the sum of all entries in the matrix $A^3$.
Matrix Representation of Row Sum Property
Let $J$ be the $3\times1$ column vector where every element is 1:
$ J = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} $The condition that the sum of entries in each row of $A$ is 1 can be represented using matrix multiplication:
$ AJ = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} a_{11} + a_{12} + a_{13} \\ a_{21} + a_{22} + a_{23} \\ a_{31} + a_{32} + a_{33} \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = J $So, we have the property $AJ = J$.
Calculating the Sum of Entries of $A^3$
The sum of all entries of any $3\times3$ matrix $M$ can be calculated as $J^T M J$, where $J^T = [1, 1, 1]$ is the transpose of $J$. We want to calculate the sum of entries for $A^3$, which is $J^T A^3 J$.
Using the property $AJ = J$, we can find $A^3 J$:
- Multiply by $A$ on the left: \( A(AJ) = AJ \) which implies \( A^2 J = J \).
- Multiply by $A$ again: \( A(A^2 J) = AJ \) which implies \( A^3 J = J \).
Final Calculation
Now, substitute $A^3 J = J$ into the expression for the sum of entries:
Sum = \( J^T A^3 J = J^T (A^3 J) = J^T J \)
Calculate $J^T J$:
$ J^T J = \begin{bmatrix} 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = (1 \times 1) + (1 \times 1) + (1 \times 1) = 1 + 1 + 1 = 3 $Therefore, the sum of all the entries of the matrix $A^3$ is 3.
Similar Questions
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
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Important Questions from Algebra
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $P = [p_{ij}]$ and $Q = [q_{ij}]$ be two square matrices of order 3 such that $q_{ij} = 2^{(i + j - 1)} p_{ij}$ and $\det(Q) = 2^{10}$. Then the value of $\det(\text{adj}(\text{adj } P))$ is:
