Problem 3 - College Algebra with Corequisite Support

Question

Can you explain whether a \(2 \times 2\) matrix with an entire row of zeros can have an inverse?

Step-by-Step Solution

Answer

No, it cannot have an inverse because its determinant is zero.

Step 1: Understand the Matrix Inverse

For a matrix to have an inverse, it must be square and its determinant must not be zero. An inverse matrix is a matrix that, when multiplied with the original, results in the identity matrix.

Step 2: Set Up the Matrix

Consider a generic 2x2 matrix with an entire row of zeros, which looks like this: \[\begin{bmatrix}a & b \0 & 0\end{bmatrix}\]

Step 3: Calculate the Determinant

The determinant of the matrix \(\begin{bmatrix} a & b \ 0 & 0 \end{bmatrix}\) is calculated as \[(ad - bc) = (a \times 0) - (b \times 0) = 0\].

Step 4: Analyze the Determinant

Since the determinant is zero, the matrix does not have an inverse. A non-zero determinant is required for a matrix to have an inverse.

Key Concepts

Determinant

The determinant is a fundamental concept in linear algebra. It's a scalar value that can be calculated from a square matrix. This value gives important information about the matrix, such as whether the matrix has an inverse. To find the determinant of a 2x2 matrix, the formula is quite simple: if you have a matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant \( \Delta \) is calculated as:\[\Delta = ad - bc\]- If the determinant is zero, the matrix does not have an inverse. This is because zero indicates that the matrix's columns are linearly dependent, meaning the vectors lie on the same line in a plane.- When the determinant is non-zero, the matrix columns are linearly independent, and the matrix is invertible.Understanding determinants is crucial to solving many matrix-related problems because they help us check the conditions for matrix operations, such as finding inverses.

Square Matrix

A square matrix is a matrix with the same number of rows and columns. Mathematically, it is expressed as an \( n \times n \) matrix. For example, the matrix \[ \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \]is a 2x2 square matrix. There are several key characteristics of square matrices:

• Square matrices of order \( n \) have \( n^2 \) elements.
• Only square matrices can have determinants, which are necessary when dealing with matrix inverses.
• They can be symmetric, diagonal, or identity matrices, each with unique properties.

Square matrices play a vital role in linear transformations. Maintaining the square shape ensures compatibility in operations such as multiplication and finding inverses, crucial for solving systems of linear equations and understanding geometric transformations.

Identity Matrix

The identity matrix is a special type of square matrix. It acts as the multiplicative identity in matrix multiplication, meaning that any matrix multiplied by an identity matrix will yield the original matrix. For a 2x2 identity matrix, it looks like this:\[ \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]Some important features of the identity matrix include:

• It is always square, with ones on the diagonal and zeros elsewhere.
• When a matrix is multiplied by its inverse, the result is the identity matrix. This property confirms that an inverse matrix has been correctly identified.
• Identity matrices are central in defining matrix inverses, as the relationship \( A \cdot A^{-1} = I \) holds.

Identity matrices function similarly to the number "1" in scalar multiplication, thus confirming their critical role in linear algebra. They assure that the structure and properties of matrices are maintained during transformations.