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Chapter 1: Problem 27

Is it possible for a singular matrix \(B\) to be row equivalent to a nonsingular matrix \(A\) ? Explain.

Short Answer

Expert verified
No, a singular matrix \(B\) cannot be row equivalent to a nonsingular matrix \(A\). This is because a singular matrix has a determinant of zero while a nonsingular matrix has a nonzero determinant. Since elementary row operations either don't change or multiply the determinant by a nonzero scalar, it is impossible for a singular matrix \(B\) to be row equivalent to a nonsingular matrix \(A\), as their determinants will always remain distinct.

Step by step solution

01

Definition of row equivalence

Two matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations. These operations are: 1. Swap two rows, 2. Multiply a row by a nonzero scalar, and 3. Add (or subtract) multiples of a row to another row.
02

Properties of singular matrices

A matrix B is said to be singular if its determinant is zero, i.e., det(B) = 0. Singular matrices don't have an inverse. When a matrix has a zero determinant, it means that the column vectors are linearly dependent, and thus, the matrix does not have full rank.
03

Properties of nonsingular matrices

A matrix A is nonsingular if its determinant is not zero, i.e., det(A) 鈮� 0. The nonsingular matrix has an inverse, meaning the column vectors are linearly independent, and the matrix has full rank.
04

Effect of row operations on determinants

Let's discuss how each elementary row operation affects a matrix determinant: 1. Swapping two rows - resulting determinant is the negative of the original determinant. 2. Multiplying a row by a nonzero scalar - resulting determinant is the original determinant multiplied by the scalar. seealso: https://math.stackexchange.com/questions/533457/2-types-of-row-operations-are-done-on-a-square-matrix-how-does-the-determinant 3. Adding (or subtracting) multiples of one row to another row - the determinant of the matrix remains unchanged.
05

Singular and nonsingular matrices row equivalence

Since the determinant of matrix A is nonzero (nonsingular) and the determinant of matrix B is zero (singular), we can conclude that no sequence of elementary row operations can transform one matrix into the other without changing their determinants. Therefore, a singular matrix B cannot be row equivalent to a nonsingular matrix A.

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Most popular questions from this chapter

Find the inverse of each of the following matrices: (a) \(\left(\begin{array}{rr}-1 & 1 \\ 1 & 0\end{array}\right)\) (b) \(\left(\begin{array}{ll}2 & 5 \\ 1 & 3\end{array}\right)\) (c) \(\left(\begin{array}{ll}2 & 6 \\ 3 & 8\end{array}\right)\) (d) \(\left(\begin{array}{ll}3 & 0 \\ 9 & 3\end{array}\right)\) (e) \(\left(\begin{array}{lll}1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1\end{array}\right)\) (f) \(\left(\begin{array}{lll}2 & 0 & 5 \\ 0 & 3 & 0 \\ 1 & 0 & 3\end{array}\right)\) (g) \(\left(\begin{array}{rrr}-1 & -3 & -3 \\ 2 & 6 & 1 \\ 3 & 8 & 3\end{array}\right)\) (h) \(\left(\begin{array}{rrr}1 & 0 & 1 \\ -1 & 1 & 1 \\ -1 & -2 & -3\end{array}\right)\)

For each of the systems of equations that follow, use Gaussian elimination to obtain an equivalent system whose coefficient matrix is in row echelon form. Indicate whether the system is consistent. If the system is consistent and involves no free variables, use back substitution to find the unique solution. If the system is consistent and there are free variables, transform it to reduced row echelon form and find all solutions $$\begin{aligned} &\text { (a) } \quad x_{1}-2 x_{2}=3\\\ &2 x_{1}-x_{2}=9 \end{aligned}$$ $$\begin{aligned} &\text { (b) } \quad 2 x_{1}-3 x_{2}=5\\\ &-4 x_{1}+6 x_{2}=8 \end{aligned}$$ $$\begin{aligned} &\text { (c) } \quad x_{1}+x_{2}=0\\\ &\begin{array}{l} 2 x_{1}+3 x_{2}=0 \\ 3 x_{1}-2 x_{2}=0 \end{array} \end{aligned}$$ $$\begin{aligned} &\text { (d) } 3 x_{1}+2 x_{2}-x_{3}=4\\\ &\begin{array}{r} x_{1}-2 x_{2}+2 x_{3}=1 \\ 11 x_{1}+2 x_{2}+x_{3}=14 \end{array} \end{aligned}$$ $$\begin{aligned} &\text { (e) } 2 x_{1}+3 x_{2}+x_{3}=1\\\ &\begin{array}{r} x_{1}+x_{2}+x_{3}=3 \\ 3 x_{1}+4 x_{2}+2 x_{3}=4 \end{array} \end{aligned}$$ $$\begin{aligned} &\text { (f) } \quad x_{1}-x_{2}+2 x_{3}=4\\\ &\begin{array}{l} 2 x_{1}+3 x_{2}-x_{3}=1 \\ 7 x_{1}+3 x_{2}+4 x_{3}=7 \end{array} \end{aligned}$$ $$\begin{aligned} &\text { (g) } x_{1}+x_{2}+x_{3}+x_{4}=0\\\ &\begin{array}{l} 2 x_{1}+3 x_{2}-x_{3}-x_{4}=2 \\ 3 x_{1}+2 x_{2}+x_{3}+x_{4}=5 \\ 3 x_{1}+6 x_{2}-x_{3}-x_{4}=4 \end{array} \end{aligned}$$ $$\begin{aligned} &\text { (h) } \quad x_{1}-2 x_{2}=3\\\ &\begin{aligned} 2 x_{1}+x_{2} &=1 \\ -5 x_{1}+8 x_{2} &=4 \end{aligned} \end{aligned}$$ $$\begin{aligned} &\text { (i) } \quad-x_{1}+2 x_{2}-x_{3}=2\\\ &\begin{aligned} -2 x_{1}+2 x_{2}+x_{3} &=4 \\ 3 x_{1}+2 x_{2}+2 x_{3} &=5 \\ -3 x_{1}+8 x_{2}+5 x_{3} &=17 \end{aligned} \end{aligned}$$ $$\begin{aligned} &\text { (j) } \quad x_{1}+2 x_{2}-3 x_{3}+x_{4}=1\\\ &\begin{array}{r} -x_{1}-x_{2}+4 x_{3}-x_{4}=6 \\ -2 x_{1}-4 x_{2}+7 x_{3}-x_{4}=1 \end{array} \end{aligned}$$ $$\begin{aligned} &\text { (k) } x_{1}+3 x_{2}+x_{3}+x_{4}=3\\\ &\begin{array}{c} 2 x_{1}-2 x_{2}+x_{3}+2 x_{4}=8 \\ x_{1}-5 x_{2}+x_{4}=5 \end{array} \end{aligned}$$ $$\begin{aligned} &\text { (I) } \quad x_{1}-3 x_{2}+\quad x_{3}=1\\\ &\begin{aligned} 2 x_{1}+x_{2}-x_{3} &=2 \\ x_{1}+4 x_{2}-2 x_{3} &=1 \\ 5 x_{1}-8 x_{2}+2 x_{3} &=5 \end{aligned} \end{aligned}$$

The matrix \\[ A=\left(\begin{array}{ll} 1 & -1 \\ 1 & -1 \end{array}\right) \\] has the property that \(A^{2}=O .\) Is it possible for a nonzero symmetric \(2 \times 2\) matrix to have this property? Prove your answer.

Let \(U\) be an \(n \times n\) upper triangular matrix with nonzero diagonal entries. (a) Explain why \(U\) must be nonsingular. (b) Explain why \(U^{-1}\) must be upper triangular.

Prove that if \(A\) is row equivalent to \(B\), then \(B\) is row equivalent to \(A\)
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