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Chapter 8: Problem 98

Determine whether the statements are true or false. All square matrices have inverses.

Short Answer

Expert verified
False, not all square matrices have inverses; some are singular with determinant zero.

Step by step solution

01

Understanding Square Matrices

A square matrix is a matrix with the same number of rows and columns. For example, a 2x2 or 3x3 matrix is square because it has equal dimensions. Generally, an n x n matrix is also referred to as a square matrix.
02

Definition of Inverse of a Matrix

A matrix has an inverse if there exists another matrix such that when the two are multiplied together, the result is the identity matrix. For a matrix A, there needs to be a matrix B such that AB = BA = I, where I is the identity matrix. Not all matrices have inverses.
03

Conditions for Matrix Inversion

A square matrix has an inverse if and only if it is non-singular. A matrix is non-singular if its determinant is not equal to zero. If the determinant is zero, the matrix is singular and does not have an inverse.
04

Counterexample

Consider a 2x2 square matrix with rows [1, 2] and [2, 4]. Calculate the determinant which is (1)(4) - (2)(2) = 4 - 4 = 0. Since the determinant is zero, this matrix is singular and does not have an inverse.
05

Conclusion

Since we've demonstrated that there are square matrices which do not have inverses (such as those with a zero determinant), we can conclude that the statement "All square matrices have inverses" is false.

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Key Concepts

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

Understanding Square Matrices
Square matrices are a fundamental concept in linear algebra, given their structural simplicity and versatility. They are matrices where the number of rows equals the number of columns. For instance, a 2x2 matrix has two rows and two columns, making it square. Similarly, any matrix with an equal count of rows and columns, like a 3x3 or 4x4, is termed as a square matrix.
Square matrices are essential because they often form the basis for discussing more complex matrix properties like determinants and eigenvalues. In linear algebra operations, square matrices are extensively used because they can be inverted, unlike rectangular matrices, provided certain conditions are met.
Matrix Inversion Conditions
Not every square matrix can be inverted, which brings us to the matrix inversion conditions. A square matrix, let's call it \( A \), has an inverse if there exists another matrix \( B \) such that the multiplication of these two matrices in either order returns the identity matrix \( I \) (i.e., \( AB = BA = I \)).
For this to be possible, \( A \) must be non-singular. Non-singular matrices have a non-zero determinant, which is a crucial factor in determining invertibility.
  • If a matrix is singular, meaning its determinant is zero, it lacks an inverse.
  • The presence of an inverse depends solely on this non-zero determinant condition.
Thus, possessing an inverse is not a universal quality among square matrices but rather conditional on these specific characteristics.
Determinant of a Matrix
The determinant of a matrix is a special number calculated from a square matrix. It provides crucial information regarding the matrix, such as whether or not the matrix can be inverted.
  • The determinant gives a measure of the matrix's ability to be reversed or "undone" through multiplication.
  • If the determinant is zero, the matrix is called singular and does not have an inverse.
  • If the determinant is not zero, the matrix is non-singular and possesses an inverse.
Let's take a simple 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \). Its determinant is calculated as \( ad - bc \). If this comes out to be zero, we immediately know that this matrix is singular and cannot be inverted. Therefore, the determinant plays a pivotal role in understanding a square matrix's properties and potential for inversion.

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Most popular questions from this chapter

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