Problem 71
Question
Let \(A=\left[\begin{array}{ccc}5 & 5 \alpha & \alpha \\ 0 & \alpha & 5 \alpha \\\ 0 & 0 & 5\end{array}\right]\), If \(\left|A^{2}\right|=25\) then \(|\alpha|\) equals (A) 5 (B) 1 (C) \(1 / 5\) (D) 5
Step-by-Step Solution
Verified Answer
The absolute value of \(\alpha\) is \(\frac{1}{5}\).
1Step 1: Identify and Use Property of Determinants
We know the property of determinants \(|A^n| = |A|^n\). Here, \(n = 2\), so \(|A^2| = |A|^2 = 25\). This simplifies to \(|A| = 5\) or \(|A| = -5\).
2Step 2: Compute the Determinant of Matrix A
Matrix \(A\) is an upper triangular matrix. The determinant of an upper triangular matrix is the product of its diagonal elements. Thus, the determinant of \(A\) is \(|A| = 5 \times \alpha \times 5 = 25\alpha\).
3Step 3: Solve the Equation for \(\alpha\)
We set \(25\alpha = 5\) or \(25\alpha = -5\) based on Step 1. Solving these gives \(\alpha = \frac{5}{25} = \frac{1}{5}\) or \(\alpha = -\frac{1}{5}\).
4Step 4: Determine \(|\alpha|\)
Since we have two solutions for \(\alpha\), \(\frac{1}{5}\) and \(-\frac{1}{5}\), the magnitude or absolute value is \(|\alpha| = \frac{1}{5}\).
Key Concepts
Matrix AlgebraProperties of DeterminantsUpper Triangular Matrix
Matrix Algebra
Matrix algebra is a branch of mathematics that deals with matrices and operations related to them. In many ways, it is similar to the algebra of numbers, but instead of working with individual numbers, matrix algebra involves working with arrays of numbers. Simple operations, such as addition and subtraction, can be applied directly between matrices of the same size. However, multiplication is a bit more complex and requires the matrix being multiplied to have specific dimensions.
Matrix algebra also includes operations such as finding the inverse, transpose, and determinant of a matrix. These operations play critical roles in advanced mathematics, statistics, and computer science. For instance, calculating the determinant is crucial in solving systems of linear equations and in understanding matrix properties.
When working with matrices, especially for more complex computations, it's important to ensure that all operations follow the rules specific to matrices. These rules often differ from regular arithmetic, which makes understanding the properties of matrices essential.
Matrix algebra also includes operations such as finding the inverse, transpose, and determinant of a matrix. These operations play critical roles in advanced mathematics, statistics, and computer science. For instance, calculating the determinant is crucial in solving systems of linear equations and in understanding matrix properties.
When working with matrices, especially for more complex computations, it's important to ensure that all operations follow the rules specific to matrices. These rules often differ from regular arithmetic, which makes understanding the properties of matrices essential.
Properties of Determinants
Determinants are a scalar value derived from a square matrix. They provide important information about the matrix, such as whether the matrix is invertible. An invertible matrix is one that has an inverse, which is crucial for solving many mathematical problems.
One key property of determinants is that they can be calculated through specific methods, one of the simplest being the multiplication of the diagonal elements in certain types of matrices, such as upper triangular matrices. The determinant can also be used to solve various puzzles and challenges within matrix algebra, such as determining the effects of matrix transformations.
Another important property is that for any square matrix, the determinant of a product of matrices equals the product of their determinants \(|AB| = |A|\.|B|\). Also, for a matrix raised to a power, the determinant of that matrix is the determinant of the original matrix raised to that power \(|A^n| = |A|^n\). These properties can simplify calculations significantly.
One key property of determinants is that they can be calculated through specific methods, one of the simplest being the multiplication of the diagonal elements in certain types of matrices, such as upper triangular matrices. The determinant can also be used to solve various puzzles and challenges within matrix algebra, such as determining the effects of matrix transformations.
Another important property is that for any square matrix, the determinant of a product of matrices equals the product of their determinants \(|AB| = |A|\.|B|\). Also, for a matrix raised to a power, the determinant of that matrix is the determinant of the original matrix raised to that power \(|A^n| = |A|^n\). These properties can simplify calculations significantly.
Upper Triangular Matrix
An upper triangular matrix is a special form of a square matrix where all the entries below the main diagonal are zero. This simple form makes certain calculations, such as determinants, much easier.
The determinant of an upper triangular matrix is simply the product of its diagonal elements. This is because the structure of the matrix allows the determinant calculation to be reduced to a straightforward multiplication, without needing any complex row operations. For example, if you have an upper triangular matrix \(A\), and its diagonal elements are \(a_{11}, a_{22},\) and \(a_{33}\), then the determinant \(|A|\) is \(|A| = a_{11} \times a_{22} \times a_{33}\).
Understanding upper triangular matrices is fundamental in linear algebra because they can be used to simplify problems and matrices via methods such as Gaussian elimination. By transforming a matrix to an upper triangular form, complex systems of equations can be resolved more straightforwardly.
The determinant of an upper triangular matrix is simply the product of its diagonal elements. This is because the structure of the matrix allows the determinant calculation to be reduced to a straightforward multiplication, without needing any complex row operations. For example, if you have an upper triangular matrix \(A\), and its diagonal elements are \(a_{11}, a_{22},\) and \(a_{33}\), then the determinant \(|A|\) is \(|A| = a_{11} \times a_{22} \times a_{33}\).
Understanding upper triangular matrices is fundamental in linear algebra because they can be used to simplify problems and matrices via methods such as Gaussian elimination. By transforming a matrix to an upper triangular form, complex systems of equations can be resolved more straightforwardly.
Other exercises in this chapter
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