Problem 47 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse of the given matrix. $$ \left[\begin{array}{llll} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

Check determinant and perform row operations; however, this matrix simplifies uniquely.

Step 1: Determine if the matrix is invertible

To find the inverse of a matrix, it must first be a square matrix and have a non-zero determinant. The given matrix is a 4x4 square matrix.

Step 2: Calculate the determinant

Calculate the determinant of the matrix to verify it's not zero. For 4x4 matrices, calculating the determinant directly can be complex, but checking various row operations denote that this matrix may not have a straightforward determinant calculation without simplification or properties.

Step 3: Check for row rank full/independent columns

Convert the matrix into a row-echelon form to check for linear independence and that each row or column contributes to a full rank. In particular, this matrix:

Step 4: Apply Gaussian elimination

Transform the given matrix to row-echelon form using Gaussian elimination to observe if you get a reduced form with all 1s in the diagonal, indicating an identity matrix when multiplied by its inverse.

Step 5: Find the inverse using row operations

Compose the matrix transformations towards the identity matrix. Since there is no row rank failure throughout this point, proceed with solving through row reduction to form an identity matrix from the original.

Step 6: Write the inverse

Upon success in forming an identity matrix using row operations, the series of operations applied help note what transforms into the adjunct matrix resultant as inverse post multipliers from identity. Ensure computations match proper inverses.

Key Concepts

Determinant of a Matrix

The concept of a determinant is fundamental when discussing whether a matrix is invertible. For a square matrix, the determinant is a scalar value that provides essential information about the matrix's properties. If the determinant of a matrix is zero, the matrix is not invertible, which means it does not have an inverse. For our given 4x4 matrix:

• The determinant helps to determine if solutions to linear equations exist.
• It checks if there are multiple solutions or a unique solution.

In practice, calculating the determinant of larger matrices manually can be cumbersome. However, shortcuts like row reduction or built-in properties can simplify this task. Remember that swapping rows of the matrix changes the sign of the determinant, and multiplying a row by a scalar multiplies the determinant by that same scalar.

Gaussian Elimination

Gaussian Elimination is a method used to simplify matrices and solve systems of linear equations. This technique systematically performs row operations to convert a matrix into a more workable form, often a row-echelon form. The key steps involve:

• Swapping rows to position a non-zero entry as a pivot.
• Scaling rows to ensure the pivot is 1.
• Adding or subtracting multiples of the pivot row from other rows to create zeros below the pivot.

This process not only assists in finding solutions to algebraic systems but can also help determine the inverse of a matrix if it leads to an identity-like matrix.

Row Echelon Form

Row Echelon Form (REF) is a simplified version of a matrix achieved by applying Gaussian Elimination. In this form:

• All non-zero rows are above any rows of all zeros.
• The leading entry of each non-zero row (the pivot) is to the right of the leading entry of the row above it.
• All entries in a column below a leading entry are zeros.

Reflecting on our procedure, converting a matrix into REF can help reveal properties like rank and linear independence, necessary for finding an inverse. Achieving a full rank (equal to the number of rows) through row reduction confirms that the matrix is invertible.

Linear Independence

Linear Independence is a vital concept when assessing the rank of a matrix. A set of vectors (or matrix rows/columns) is considered linearly independent if no vector can be expressed as a linear combination of the others. In simpler terms:

• If vectors are independent, each vector adds new information, impacting the span.
• Dependent vectors can be omitted since they are just combinations of others in the set.

For a matrix to be invertible, its rows (and columns) must be linearly independent. Checking for linear independence often involves transforming the matrix to row echelon form, ensuring that each row adds to the rank, confirming a non-zero determinant, and thus, invincibility.