Let A=(ai j) be a square matrix of order n such that ai j=0 if i<j . Show that |A|=a11 a22 ⋯an n

step 1 Given in this question is A equals to Aij, B a square matrix of order N, such that Aij is equal

Let A=(ai j) be a square matrix of order n such that ai j=0...

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Key Concepts

Triangular Matrix

A triangular matrix is a type of square matrix where all the entries either above or below the main diagonal are zero. This structure greatly simplifies many matrix operations, particularly the calculation of determinants since many terms in the expansion automatically become zero.

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix and provides important information about the matrix, such as whether it is invertible. It also represents the scaling factor applied to volumes under the linear transformation defined by the matrix.

Product of Diagonal Elements

In the case of triangular matrices, the determinant is equal to the product of the entries on the main diagonal. This is because, in the permutation expansion of the determinant, all non-diagonal terms involve zero elements due to the triangular structure, leaving only the product of the diagonal entries.

Mathematical Induction

Mathematical induction is a proof technique that verifies the validity of a statement for all natural numbers by first proving it for a base case and then showing that if the statement holds for an arbitrary case, it also holds for the subsequent case. It is commonly used to prove properties like the determinant of a triangular matrix.

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