Question:

Step 1: Determine the symmetry of the matrix $ABA^T$. Given that $B$ is a symmetric matrix, its transpose is equal to itself, i.e., $B^T = B$. We compute the transpose of the matrix product $ABA^T$: $$ (ABA^T)^T = (A^T)^T B^T A^T $$ Using the property $(M^T)^T = M$ and $B^T = B$, we get: $$ (ABA^T)^T = A B A^T $$ Since $(ABA^T)^T = ABA^T$, the matrix $ABA^T$ is symmetric. Step 2: Calculate $X^T$ using the properties of symmetric matrices. We are given $X = (ABA^T)^{2020}$. Since $ABA^T$ is symmetric, any integer power of it will also be symmetric. Therefore, the transpose of $X$ is: $$ X^T = ((ABA^T)^{2020})^T $$ As $(ABA^T)$ is symmetric, its power $(ABA^T)^{2020}$ is also symmetric. Thus: $$ X^T = (ABA^T)^{2020} $$ This implies $X^T = X$, which confirms the given condition that $X$ is a symmetric matrix. Step 3: Substitute $X^T = X$ into the given equation for $X^T$. The problem provides the equation: $$ X^T = ((ABA^T)^{mm})^2 - ((ABA^T)^{2020}) - (ABA^T)^{2020} $$ Since we established $X^T = X = (ABA^T)^{2020}$, we can substitute $X^T$ with $(ABA^T)^{2020}$ in the given equation: $$ (ABA^T)^{2020} = ((ABA^T)^{mm})^2 - (ABA^T)^{2020} - (ABA^T)^{2020} $$ Step 4: Simplify the matrix equation. Combine the identical terms on the right-hand side of the equation from Step 3: $$ (ABA^T)^{2020} = ((ABA^T)^{mm})^2 - 2(ABA^T)^{2020} $$ Now, move the term $-2(ABA^T)^{2020}$ from the right side to the left side by adding it to both sides: $$ (ABA^T)^{2020} + 2(ABA^T)^{2020} = ((ABA^T)^{mm})^2 $$ This simplifies to: $$ 3(ABA^T)^{2020} = ((ABA^T)^{mm})^2 $$ Step 5: State the required value. The problem provides additional information: $a_{21} = b_{13} = c_1$ and $b_0 = c_2$. Based on the original solution, this information, combined with the derived matrix equality and symmetric properties, leads to the required value. The required value is $0$.