1) a Consider the following matrix: [5 3 3 A = 3 5 3 3 3 5 il Verify that the two following vectors are the eigenvectors of this matrix without solving the characteristic equation. u2 State the corresponding eigenvalues of each vector clearly without solving the charac- teristic equation. Consider the square matrices given below. Find their eigenvalues by using eigenvalue prop- erties of matrices (do not solve the characteristic equation). State clearly which property you used to find these eigenvalues. 1 0 3 4] 0 2 3 5 i) 0 0 3 9 0 0 0 4 [1 0 0 0] 0 2 0 0 0 0 3 0 0 0 0 4 i1)
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
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