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Chapter 6: Q3E (page 308)

The determinant of any diagonaln脳nmatrix is the product of its diagonal entries.

Short Answer

Expert verified

Yes, the determinant of a diagonal matrix is the product of elements of its diagonal. So, the given condition is true.

Step by step solution

01

Definition. 

A determinant is a unique number associatedwith a square matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

02

To check whether the given condition is true or false.

Using the very definition of the determinant, we can see that this equation applies for any diagonaln脳n matrix.

The determinant of a lower triangular matrix (or an upper triangular matrix) is the product of the diagonal entries.

Therefore, the determinant of a diagonal matrix is the product of elements of its diagonal.

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Most popular questions from this chapter

If all the diagonal entries of an n脳nmatrix Aare odd integers and all the other entries are even integers, thenA must be an invertible matrix.

(For those who have studied multivariable calculus.) Let Tbe an invertible linear transformation from鈩�2to鈩�2, represented by the matrix M. Let惟1be the unit square in 鈩�2and惟2its image under T . Consider a continuous functionf(x,y)from鈩�2to鈩�, and define the functiong(u,v)=f(T(u,v)). What is the relationship between the following two double integrals?

鈭�惟2f(x,y)dAand鈭�惟1g(u,v)dA

Your answer will involve the matrix M. Hint: What happens whenf(x,y)=1, for allx,y?

There exist real invertible 3脳3 matrices A andSsuch that STAS=-A

Explain why any patternPin a matrixA, other than the diagonal pattern, contains at least one entry below the diagonal and at least one entry above the diagonal.

The determinant of all orthogonal matrices is 1 .
See all solutions

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