1. Key Concepts and Formulas

This problem requires a strong understanding of matrix operations, especially matrix multiplication and addition, combined with the ability to identify patterns and apply summation formulas.

1. Matrix Multiplication: To find powers of a matrix ($A^n$), we repeatedly multiply the matrix by itself. For two matrices $A = [a_{ij}]$ and $B = [b_{jk}]$, their product $C = AB$ has elements $c_{ik} = \sum_j a_{ij} b_{jk}$. This operation is performed row by column.
2. Matrix Addition: When summing matrices (like $M = A + A^2 + \dots + A^{20}$), corresponding elements are added. If $C = A+B$, then $c_{ij} = a_{ij} + b_{ij}$.
3. Pattern Recognition: For powers of matrices, especially upper or lower triangular matrices, elements often follow arithmetic or polynomial sequences. Identifying these patterns is crucial to generalize $A^n$.
4. Summation Formulas: We will use the following standard formulas for sums of series:
   • Sum of the first $N$ natural numbers: $\sum_{k=1}^{N} k = \frac{N(N+1)}{2}$
   • Sum of the first $N$ squares: $\sum_{k=1}^{N} k^2 = \frac{N(N+1)(2N+1)}{6}$

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2. Step 1: Analyze the Structure of Matrix A and its Powers ($A^k$)

Our first step is to compute the first few powers of $A$ to identify a pattern. This is a standard strategy for problems involving sums of matrix powers, as directly summing 20 matrices is impractical.

Given matrix $A$: A = \left[ {\matrix{ 1 & 1 & 1 \cr 0 & 1 & 1 \cr 0 & 0 & 1 \cr } } \right]

Let's calculate $A^2$: A^2 = A \cdot A = \left[ {\matrix{ 1 & 1 & 1 \cr 0 & 1 & 1 \cr 0 & 0 & 1 \cr } } \right] \left[ {\matrix{ 1 & 1 & 1 \cr 0 & 1 & 1 \cr 0 & 0 & 1 \cr } } \right] = \left[ {\matrix{ (1 \cdot 1 + 1 \cdot 0 + 1 \cdot 0) & (1 \cdot 1 + 1 \cdot 1 + 1 \cdot 0) & (1 \cdot 1 + 1 \cdot 1 + 1 \cdot 1) \cr (0 \cdot 1 + 1 \cdot 0 + 1 \cdot 0) & (0 \cdot 1 + 1 \cdot 1 + 1 \cdot 0) & (0 \cdot 1 + 1 \cdot 1 + 1 \cdot 1) \cr (0 \cdot 1 + 0 \cdot 0 + 1 \cdot 0) & (0 \cdot 1 + 0 \cdot 1 + 1 \cdot 0) & (0 \cdot 1 + 0 \cdot 1 + 1 \cdot 1) \cr } } \right] = \left[ {\matrix{ 1 & 2 & 3 \cr 0 & 1 & 2 \cr 0 & 0 & 1 \cr } } \right]

Next, let's calculate $A^3$: A^3 = A^2 \cdot A = \left[ {\matrix{ 1 & 2 & 3 \cr 0 & 1 & 2 \cr 0 & 0 & 1 \cr } } \right] \left[ {\matrix{ 1 & 1 & 1 \cr 0 & 1 & 1 \cr 0 & 0 & 1 \cr } } \right] = \left[ {\matrix{ (1 \cdot 1 + 2 \cdot 0 + 3 \cdot 0) & (1 \cdot 1 + 2 \cdot 1 + 3 \cdot 0) & (1 \cdot 1 + 2 \cdot 1 + 3 \cdot 1) \cr (0 \cdot 1 + 1 \cdot 0 + 2 \cdot 0) & (0 \cdot 1 + 1 \cdot 1 + 2 \cdot 0) & (0 \cdot 1 + 1 \cdot 1 + 2 \cdot 1) \cr (0 \cdot 1 + 0 \cdot 0 + 1 \cdot 0) & (0 \cdot 1 + 0 \cdot 1 + 1 \cdot 0) & (0 \cdot 1 + 0 \cdot 1 + 1 \cdot 1) \cr } } \right] = \left[ {\matrix{ 1 & 3 & 6 \cr 0 & 1 & 3 \cr 0 & 0 & 1 \cr } } \right]

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3. Step 2: Determine the General Form of $A^k$

By observing the pattern in $A^1$, $A^2$, and $A^3$: A^1 = \left[ {\matrix{ 1 & 1 & 1 \cr 0 & 1 & 1 \cr 0 & 0 & 1 \cr } } \right] A^2 = \left[ {\matrix{ 1 & 2 & 3 \cr 0 & 1 & 2 \cr 0 & 0 & 1 \cr } } \right] A^3 = \left[ {\matrix{ 1 & 3 & 6 \cr 0 & 1 & 3 \cr 0 & 0 & 1 \cr } } \right]

We can deduce the general form for $A^k$:

• The diagonal elements ($a_{11}$, $a_{22}$, $a_{33}$) are always 1.
• The elements $a_{12}$ and $a_{23}$ are equal to the power $k$. (e.g., $A^1 \to 1$, $A^2 \to 2$, $A^3 \to 3$).
• The element $a_{13}$ is the sum of the first $k$ natural numbers. (e.g., $A^1 \to 1$, $A^2 \to 1+2=3$, $A^3 \to 1+2+3=6$). This is given by the formula $\frac{k(k+1)}{2}$.
• The elements below the main diagonal ($a_{21}$, $a_{31}$, $a_{32}$) are always 0, as $A$ is an upper triangular matrix, and powers of upper triangular matrices remain upper triangular.

Thus, the general form for $A^k$ is: A^k = \left[ {\matrix{ 1 & k & \frac{k(k+1)}{2} \cr 0 & 1 & k \cr 0 & 0 & 1 \cr } } \right]

Self-check/Alternative Method (Optional but good for understanding): We could also express $A$ as $I+N$, where $I$ is the identity matrix and N = \left[ {\matrix{ 0 & 1 & 1 \cr 0 & 0 & 1 \cr 0 & 0 & 0 \cr } } \right]. Note that N^2 = \left[ {\matrix{ 0 & 0 & 1 \cr 0 & 0 & 0 \cr 0 & 0 & 0 \cr } } \right] and $N^3 = 0$. Using the binomial expansion for $(I+N)^k = I^k + \binom{k}{1}I^{k-1}N + \binom{k}{2}I^{k-2}N^2 + \binom{k}{3}I^{k-3}N^3 + \dots$: $A^k = I + kN + \frac{k(k-1)}{2}N^2$ (since $N^3=0$ and higher powers are also zero). Substituting $I$, $N$, and $N^2$: A^k = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right] + k \left[ {\matrix{ 0 & 1 & 1 \cr 0 & 0 & 1 \cr 0 & 0 & 0 \cr } } \right] + \frac{k(k-1)}{2} \left[ {\matrix{ 0 & 0 & 1 \cr 0 & 0 & 0 \cr 0 & 0 & 0 \cr } } \right] A^k = \left[ {\matrix{ 1 & k & k + \frac{k(k-1)}{2} \cr 0 & 1 & k \cr 0 & 0 & 1 \cr } } \right] = \left[ {\matrix{ 1 & k & \frac{2k + k^2 - k}{2} \cr 0 & 1 & k \cr 0 & 0 & 1 \cr } } \right] = \left[ {\matrix{ 1 & k & \frac{k(k+1)}{2} \cr 0 & 1 & k \cr 0 & 0 & 1 \cr } } \right] This confirms our derived pattern for $A^k$.

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4. Step 3: Calculate the Matrix M = A + A^2 + ... + A^20

The matrix $M$ is the sum of $A^k$ for $k$ from 1 to 20. According to matrix addition rules, each element of $M$ is the sum of the corresponding elements of $A^k$. Let $m_{ij}$ denote the element in the $i$-th row and $j$-th column of $M$.

We will sum each element position from $k=1$ to $k=20$:

• Element $m_{11}$ (and $m_{22}$, $m_{33}$): These are always 1 in $A^k$. $m_{11} = \sum_{k=1}^{20} (A^k)_{11} = \sum_{k=1}^{20} 1 = 20 \cdot 1 = 20$ Similarly, $m_{22} = 20$ and $m_{33} = 20$.

• Element $m_{12}$ (and $m_{23}$): These are $k$ in $A^k$. $m_{12} = \sum_{k=1}^{20} (A^k)_{12} = \sum_{k=1}^{20} k = \frac{20(20+1)}{2} = \frac{20 \cdot 21}{2} = 10 \cdot 21 = 210$ Similarly, $m_{23} = 210$.

• Element $m_{13}$: This is $\frac{k(k+1)}{2}$ in $A^k$. $m_{13} = \sum_{k=1}^{20} (A^k)_{13} = \sum_{k=1}^{20} \frac{k(k+1)}{2} = \frac{1}{2} \sum_{k=1}^{20} (k^2+k)$ Using the summation formulas: $m_{13} = \frac{1}{2} \left( \sum_{k=1}^{20} k^2 + \sum_{k=1}^{20} k \right)$ We know $\sum_{k=1}^{20} k = 210$. For $\sum_{k=1}^{20} k^2$: $\sum_{k=1}^{20} k^2 = \frac{20(20+1)(2 \cdot 20 + 1)}{6} = \frac{20 \cdot 21 \cdot 41}{6} = 10 \cdot 7 \cdot 41 = 70 \cdot 41 = 2870$ So, $m_{13} = \frac{1}{2} (2870 + 210) = \frac{1}{2} (3080) = 1540$

• Elements $m_{21}$, $m_{31}$, $m_{32}$: These are always 0 in $A^k$. $m_{21} = \sum_{k=1}^{20} 0 = 0$ Similarly, $m_{31} = 0$ and $m_{32} = 0$.

Combining these, the matrix $M$ is: M = \left[ {\matrix{ 20 & 210 & 1540 \cr 0 & 20 & 210 \cr 0 & 0 & 20 \cr } } \right]

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5. Step 4: Calculate the Sum of All Elements of M

Finally, we sum all the elements of the matrix $M$: Sum of all elements of $M = m_{11} + m_{12} + m_{13} + m_{21} + m_{22} + m_{23} + m_{31} + m_{32} + m_{33}$ Sum $= 20 + 210 + 1540 + 0 + 20 + 210 + 0 + 0 + 20$ Sum $= (20 + 20 + 20) + (210 + 210) + 1540$ Sum $= 60 + 420 + 1540$ Sum $= 480 + 1540$ Sum $= 2020$

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Tips and Common Mistakes:

• Careful Matrix Multiplication: Always double-check your matrix multiplications, especially for the $(1,3)$ element, as it often involves more terms. A single error here can propagate through the entire solution.
• Pattern Recognition is Key: For problems involving powers of matrices, computing the first few powers and identifying a pattern is almost always the intended method. Don't try to compute all 20 matrices individually!
• Correct Summation Formulas: Ensure you use the correct summation formulas for arithmetic series ($\sum k$) and series of squares ($\sum k^2$).
• Upper Triangular Matrix Property: Remember that powers of an upper triangular matrix remain upper triangular. This means all elements below the main diagonal will remain zero, simplifying calculations.
• Binomial Expansion for $A=I+N$: For matrices like $A=I+N$ where $N$ is nilpotent (i.e., $N^k=0$ for some $k$), the binomial expansion $(I+N)^k$ is a very efficient way to find $A^k$.

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Summary and Key Takeaway:

This problem demonstrates a common strategy for handling sums of matrix powers:

1. Deconstruct the matrix: If possible, write the matrix $A$ as $I+N$ where $N$ is a simpler matrix (often nilpotent).
2. Find the general form of $A^k$: Use pattern recognition from the first few powers or binomial expansion to derive a formula for the elements of $A^k$.
3. Sum element-wise: For the sum matrix $M$, sum the corresponding elements of $A^k$ over the given range, applying relevant summation formulas.
4. Final calculation: Perform the required operation (e.g., sum of all elements, determinant) on the resulting matrix $M$.

The final answer is $\boxed{2020}$.