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Chapter 8: Problem 69

Determine whether the statement is true or false. Justify your answer. If a square matrix has an entire row of zeros, then the determinant will always be zero.

Short Answer

Expert verified
The statement is true. If a square matrix has a row or column of zeros, its determinant will indeed always be zero.

Step by step solution

01

Understand the statement

The statement claims that if a square matrix has an entire row of zeros, the determinant of that matrix will always be zero.
02

Recall determinant calculation

Determinant of a square matrix is usually calculated by selecting a row or a column and making use of the 'minus-plus' rule which is also referred as cofactor expansion. When there's a row or a column of zeros, the process simplifies, as multiplying anything by zero results in zero.
03

Evaluate the statement

Now if we look at the statement, if any row of a square matrix is full of zeros, all the terms in that determinant calculation will be zero since they're multiplied with zero according to the rule of determinant calculation. So, it results in the determinant being zero.

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Most popular questions from this chapter

(A) find the determinant of \(A,\) (b) find \(A^{-1},\) (c) find \(\operatorname{det}\left(A^{-1}\right),\) and (d) compare your results from parts (a) and (c). Make a conjecture based on your results. $$A=\left[\begin{array}{ll} 5 & -1 \\ 2 & -1 \end{array}\right]$$

(A) find the determinant of \(A,\) (b) find \(A^{-1},\) (c) find \(\operatorname{det}\left(A^{-1}\right),\) and (d) compare your results from parts (a) and (c). Make a conjecture based on your results. $$A=\left[\begin{array}{rr} 1 & 2 \\ -2 & 2 \end{array}\right]$$

Solve for \(x\) $$\left|\begin{array}{rr} x & 4 \\ -1 & x \end{array}\right|=20$$

Create a system of linear equations in two variables that has the solution (2,-1) as its only solution. (There are many correct answers.)

The sums have been evaluated. Solve the given system for \(a\) and \(b\) to find the least squares regression line for the points. Use a graphing utility to confirm the results. $$\left\\{\begin{aligned} 6 b+15 a &=23.6 \\ 15 b+55 a &=48.8 \end{aligned}\right.$$
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