Let a be a symmetric tridiagonal matrix ie a is symmetric...

Let A be a symmetric tridiagonal matrix (i.e., A is symmetric and aij = 0 whenever |i-j| > 1). Let B be the matrix formed from A by deleting the first two rows and columns. Show that det(A) = a11 det(M11) - a12^2 det(B).

Transcript

00:01 Hi, you know what the given question be a given that a is a symmetric tridiagonal matrix and further here we are given that for this we can say that a ij is equal to 0 whenever modulus of i minus j is greater than 1 now let b be the matrix formed from a by deleting two rows and a column so b is a matrix obtained by deleting two row and column from a now here we need to show that determinant of a is equal to a 1 1 multiplied with determinant of m 1 1 minus a 1 2 square multiplied with determinant of p so here in our case for the tridiagonal matrix we know that determinant of a can be written as a 1 1 a 1 2 a 2 1 a 2 2 a 2 3 again here we have a 3 2 and this number will continue this will continue further and this will continue so here this will continue up to a n n minus 1 this will continue up to a n 1 into n and this will continue up to a n n so this is the value of the determinant of tridiagonal matrix now further we know that here if we take a 1 1 as a common so this can be written as a 1 1 which can be written as a 2 2 a 2 3 again a 3 2 a 3 3 a 3 4 further this value is to be continued up to a n n this value is to be continued up to a n minus 1 times of n and further this value is continued up to a n m minus 1 so here this is the determinant minus further if we now subtract this value.

02:33 This can be written as a 1 2 minus a 2 1 a 2 3 again a 3 3 a 4 3 here we have a 3 4 and further this values are to be continued up to a n minus 1 n this value is continued up to a n n and this value is continued up to a n m minus 1 so this is the value now further this value can be again simplified as we know that a 1 2 is equal to a 2 1 for the symmetric matrix then again, if we take out this value and the determinant can be written as a 1 1 into here we will call this matrix as minor of 1 1 and if we remove this matrix, then the resultant matrix will be minor of m 1 1 so determinant of m 1 1 minus a 1 2 square as we have taken this out common twice multiplied with m 1 1 so here now we know that the matrix obtained by deleting a rows and column can be written as determinant of b so here by rewriting this value we can say that determinant of a is equal to a 1 1 multiplied with determinant of m 1 1 and the new matrix will be b matrix so a 1 2 square multiplied with determinant of b...

Question

Let A be a symmetric tridiagonal matrix (i.e., A is symmetric and aij = 0 whenever |i-j| > 1). Let B be the matrix formed from A by deleting the first two rows and columns. Show that det(A) = a11 det(M11) - a12^2 det(B).

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