Number of Ways to Multiply a Chain of M Matrices understanding the Problem: When multiplying a chain of matrices, the order in which we perform the multiplications can significantly impact the number of scalar multiplications required. This is because matrix multiplication is not commutative. The Catalan Number Connection: The number of ways to parenthesize a chain of M matrices is given by the Catalan number C(M-1) Catalan Numbers: Catalan numbers are a sequence of natural numbers that occur in many counting problems. The nth Catalan number, denoted as C_n, can be calculated using the following formula: C_n (1/(n+1)) (2n choose n) where (2n choose n) is a binomial coefficient. Why Catalan Numbers? Each way to parenthesize a chain of matrices corresponds to a unique way to divide a convex polygon with M+1 sides into triangles using non-intersecting diagonals. This is a classic combinatorial problem solved by Catalan numbers. Example: Consider a chain of 4 matrices: A, B, C, and D. We can parenthesize them in the following ways: ((ABC)D (ABC))D ABC)D) A(BICD)) (ABCD) There are 5 ways to parenthesize this chain, which corresponds to C 3 - 5. In Summary: The number of ways to multiply a chain of M matrices is given by the (M-1)th Catalan number, C(M-1). This number grows rapidly with M, highlighting the importance of finding the optimal parenthesization to minimize the number of scalar multiplications.

Write the functions that will calculate the trace and the determinant of the matrices. The trace of the matric is defined as the sum of the diagonal elements, i.e., Tr = ₁ aii. The standard formula to find the determinant of a 3 x 3 matrix is to a break it down into smaller 2 x 2 determinants Mij, known as minors, det The minor Mij is defined to be the determinant of the (n-1) x (n-1) matrix that results from A by removing the i-th row and the j-th column. The minors would be determinants of the 2 × 2 matrices, which could be easliy calculated from the formula n (A)=(-1)+c j=1 b deta d C ¡Mij. aij = ad bc.

True or False The inverse of a matrix M can be calculated in Python with the command Inverse=np.linalg.inv(M)

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Write pseudo-code algorithms for matrix addition, scalar multiplication, and product of two matrices in 3*3 matrix.

Calculate all matrices of order two such that: (a) its square is the null matrix; A² = A (b) are independent, that is, **Use python coding**

Do

Standard matrix multiplication method works ……… the Stassen’s algorithm of  matrix multiplication

MatrixMultiplication algorithm (A1, A2, ... , An)pre-cond: An instance is a series of n matrices.post-cond: optSol is the bracketing that takes the fewest multiplications, and optCost is the resultant number of multiplications.

You have to run Prim's algorithm for the problem defined by adjacency matrix: 1 2 3 4 5 6 7 8 9 1 0 10 9 999 999 17 999 999 999 2 10 10 3 9 11 0 14 4 2 999 999 13 999 14 0 7 999 999 999 999 999 4 999 4 7 0 999 2 8 999 999 567 999 2 999 999 0 6 999 1 999 17 999 999 2 6 0 999 7 999 999 999 999 8 999 999 0 11 4 8 999 13 999 999 1 7 11 0 8 9 999 999 999 999 999 999 4 8 0 1. We started from the vertex vl, so initially we have Y = {v1}: initial nearest 1 2 3 4 5 6 7 8 9 16 1 1 1 1 1 1 1 1 distance -1 10 9 999 999 17 999 999 999 Print out the values stored in the nearest and distance arrays after first iteration of Prim's algorithm. Specify the value of vnear and the next vertex that has to be added to Y Hint: use (copy) the table above to record your answer.