Let A denote an upper triangular matrix and |A| ≠0. Statement 1 A^-1 is also an upper triangular

Let A denote an upper triangular matrix and |A| ≠0....

Let A denote an upper triangular matrix and $|\mathrm{A}| \neq 0$.
Statement 1
$\mathrm{A}^{-1}$ is also an upper triangular matrix.
and
Statement 2
Product of two upper triangular matrices of the same order is also an upper triangular matrix.

Key Concepts

Matrix Multiplication for Triangular Matrices

Multiplying two upper triangular matrices results in another matrix that is also upper triangular. This property comes from the fact that the multiplication process involves summing products that, due to the zero entries below the main diagonal in the factors, do not produce any nonzero entries below the main diagonal in the product matrix.

Upper Triangular Matrix

An upper triangular matrix is a square matrix in which all the entries below the main diagonal are zero. This property simplifies many matrix operations, such as solving systems of linear equations, computing determinants, and eigenvalue analysis, because many calculations involve sums or products that are simplified by the zero entries.

Matrix Inversion in the Context of Triangular Matrices

For a triangular matrix with nonzero entries on the diagonal—which ensures that its determinant is nonzero—an inverse exists and retains the triangular form. This occurs because the structure of the triangular matrix allows methods such as back-substitution to compute the inverse, and the resulting matrix does not introduce nonzero values below the diagonal.

Transcript

00:01 So in the given question we have two statements that are given and we have in the question it is first said that a is an upper triangular matrix, a is an upper triangular matrix and the determinant of a is not equal to 0.

00:25 So this is said in the question and in the first statement it is told that the inverse of the matrix a is also an upper triangular matrix, upper triangular matrix.

00:45 And in the statement two given in the question we are told that the product of two upper triangular matrices of the same order is also an upper triangular matrix.

00:58 The product of two upper triangular matrices is also an upper triangular matrix.

01:25 So these are the two statements, right? so we should know what is an upper triangular matrix first.

01:32 So in an upper triangular matrix the elements that are not in the diagonal elements, the diagonals that are under the main diagonal are all zeros, right? so the elements under the main diagonal are all zero and we call such a matrix and an upper triangular matrix since we have only elements in the upper triangle of the matrix, right? so this is an example of an upper triangular matrix and now what we can do is we can see that there are two statements.

02:15 So a property of an upper triangular matrix is that if a is upper triangular, if a is upper triangular and the determinant of a is not equal to then a inverse is also an upper triangular matrix, an upper triangular matrix, right? so this is a property of an upper triangular matrix and since we have this property we can say that statement one, statement one is true, right? now we have the statement two, right? and in statement 2 we are told that the product of two upper triangular matrices is also an upper triangular matrix, which is again another property of the which is again another property of an upper triangular matrix...

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