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Chapter 3: Problem 4

Determine which property of determinants the equation illustrates. $$\left|\begin{array}{rrr}-3 & 2 & 1 \\\6 & 0 & 0 \\\\-3 & 2 & 1\end{array}\right|=0$$

Short Answer

Expert verified
The given equation illustrates the property of determinants that the determinant of a matrix with two identical rows (or columns) is zero.

Step by step solution

01

Observation

Observe that the first and third rows of the given matrix are identical.
02

Determine the property

The determinant of a square matrix with two identical rows is zero, which is a property of determinants. Since the given matrix has two identical rows, the determinant of the matrix is zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Properties of Determinants
The properties of determinants are foundational elements when dealing with matrices, especially square matrices. Determinants are special numbers that can be calculated from a square matrix. They provide important information about the matrix, such as whether it is invertible. Let鈥檚 dive into some of the key properties:
  • Identical Rows or Columns: If a square matrix contains two identical rows or columns, its determinant is zero. This property is crucial for the problem at hand, as it directly relates to identifying similarities in matrix rows.

  • Linear Combination: The determinant does not change if a row is replaced by its linear combination with other rows.

  • Row Exchange: Swapping two rows (or columns) of a matrix results in changing the sign of the determinant.

  • Zero Row or Column: If any row or column of a matrix consists entirely of zeros, the determinant is zero.

  • Upper and Lower Triangular Matrices: For these matrices, the determinant is the product of the diagonal elements.
Identical Rows in a Matrix
Identical rows in a matrix have a significant impact when calculating determinants. When a matrix is said to have identical rows, it means that two or more rows of the matrix are exactly the same. Recognizing this feature is critical:
  • Determinant Equals Zero: A fundamental property of determinants is that if any two rows of a square matrix are identical, the determinant of that matrix will be zero. This is because the matrix is dependent on these rows, causing a reduction in its rank.

  • Checking for Identical Rows: When dealing with matrices, always check for identical rows early in your calculations as this can simplify calculations significantly.
Identical rows can make the entire matrix's solution approach much easier since knowing that the determinant is zero can save computation time.
Square Matrix
A square matrix is a matrix that has the same number of rows and columns. This specific configuration allows for special properties like determinants to be calculated. Understanding square matrices is essential when studying linear algebra:
  • Dimension: The dimension of a square matrix is defined by the number of rows (or columns) it contains, typically denoted as an \(n \times n\) matrix.

  • Calculation of Determinants: Only square matrices have determinants, a critical aspect that indicates whether a matrix is singular (determinant shows zero) or non-singular (non-zero determinant).

  • Identity Matrix: Within square matrices, the identity matrix serves as a multiplicative identity, much like 1 does for real numbers. For a given size \(n\), it is an \(n \times n\) matrix with ones on the diagonal and zeros elsewhere.
Understanding these aspects of square matrices provides the foundation for exploring more complex concepts in linear algebra.

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Most popular questions from this chapter

Determine whether the matrix is orthogonal. An invertible square matrix \(A\) is orthogonal when \(A^{-1}=A^{T}\). $$\left[\begin{array}{rrr}1 / \sqrt{2} & 0 & -1 / \sqrt{2} \\\0 & 1 & 0 \\\1 / \sqrt{2} & 0 & 1 / \sqrt{2}\end{array}\right]$$

Use the determinant of the coefficient matrix to determine whether the system of linear equations has a unique solution. $$\begin{aligned}x_{1}-x_{2}+x_{3} &=4 \\\2 x_{1}-x_{2}+x_{3} &=6 \\\3 x_{1}-2 x_{2}+2 x_{3} &=0\end{aligned}$$

Use the determinant of the coefficient matrix to determine whether the system of linear equations has a unique solution. $$\begin{aligned}&3 x_{1}-4 x_{2}=2\\\&\frac{2}{3} x_{1}-\frac{8}{9} x_{2}=1\end{aligned}$$

Find (a) \(\left|\boldsymbol{A}^{T}\right|,(\mathbf{b})\left|\boldsymbol{A}^{2}\right|,(\mathbf{c})\left|\boldsymbol{A A}^{T}\right|,(\mathbf{d})|\boldsymbol{2} \boldsymbol{A}|,\) and \((\mathbf{e})\left|\boldsymbol{A}^{-\mathbf{1}}\right|\). $$A=\left[\begin{array}{rrr}5 & 0 & 0 \\\1 & -3 & 0 \\\0 & -1 & 2\end{array}\right]$$

Prove that the determinant of an invertible matrix \(A\) is equal to \(\pm 1\) when all of the entries of \(A\) and \(A^{-1}\) are integers. Getting Started: Denote det( \(A\) ) as \(x\) and \(\operatorname{det}\left(A^{-1}\right)\) as \(y\) Note that \(x\) and \(y\) are real numbers. To prove that \(\operatorname{det}(A)\) is equal to \(\pm 1,\) you must show that both \(x\) and \(y\) are integers such that their product \(x y\) is equal to 1 (i) Use the property for the determinant of a matrix product to show that \(x y=1\) (ii) Use the definition of a determinant and the fact that the entries of \(A\) and \(A^{-1}\) are integers to show that both \(x=\operatorname{det}(A)\) and \(y=\operatorname{det}\left(A^{-1}\right)\) are integers. (iii) Conclude that \(x=\operatorname{det}(A)\) must be either 1 or \(-1\) because these are the only integer solutions to the equation \(x y=1\)
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