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Chapter 8: Problem 70

Determine whether the statement is true or false. Justify your answer. If two columns of a square matrix are the same, then the determinant of the matrix will be zero.

Short Answer

Expert verified
The statement is true. If two columns of a square matrix are the same, then the determinant of the matrix is indeed zero according to the properties of determinants.

Step by step solution

01

Understanding the statement

The statement suggests that if in a square matrix, two columns are the same, the determinant of that matrix would be zero.
02

Addressing the concept of determinant

The determinant is a special number that can be calculated from a matrix. One key property is that the determinant of a matrix changes its sign when two rows (or equivalently columns) are interchanged. This is called the property of alternate changes.
03

Applying the alternate property

Another version of the alternate property says that the determinant is zero if two rows (or equivalently columns) in the matrix are equal. This means that if you switch the two identical columns, the determinant of the matrix stays the same because switching two identical columns doesn't change the matrix. But because of the property of alternate changes, the determinant should change its sign. The only way for a number to be equal to its negative is for that number to be zero. Thus, the determinant of the matrix must be zero if two columns are equal.

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Most popular questions from this chapter

Use matrices to solve the system of equations, if possible. Use Gaussian elimination with back-substitution. $$\left\\{\begin{array}{l} -x+y=-22 \\ 3 x+4 y=4 \\ 4 x-8 y=32 \end{array}\right.$$

Use matrices to solve the system of equations, if possible. Use Gaussian elimination with back-substitution. $$\left\\{\begin{array}{l} 2 x+6 y=16 \\ 2 x+3 y=7 \end{array}\right.$$

(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. $$A=\left[\begin{array}{rrr} 1 & -3 & -2 \\ -1 & 3 & 1 \\ 0 & 2 & -2 \end{array}\right]$$

Use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{c} 2 x+5 y+w=11 \\ x+4 y+2 z-2 w=-7 \\ 2 x-2 y+5 z+w=3 \\ x-3 w=-1 \end{array}\right.$$

Determine whether the statement is true or false. Justify your answer. Multiplication of an invertible matrix and its inverse is commutative.
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