Question

Show that \(B\) is the inverse of \(A\). \(A=\left[\begin{array}{rrrr}-1 & 1 & 0 & -1 \\ 1 & -1 & 2 & 0 \\ -1 & 1 & 2 & 0 \\ 0 & -1 & 1 & 1\end{array}\right]\) \(B=\frac{1}{4}\left[\begin{array}{rrrr}-4 & 1 & 1 & -4 \\ -4 & -1 & 3 & -4 \\\ 0 & 1 & 1 & 0 \\ -4 & -2 & 2 & 0\end{array}\right]\)

Short answer

By performing the multiplication of matrices \(A\) and \(B\), and observing that the result is the identity matrix, it is confirmed that \(B\) is the inverse of \(A\) which validates the given equation.

Step-by-step solution

Set up the multiplication

The first step is to write out the multiplication of the two matrices \(A\) and \(B\).

Perform the multiplication

For each element in the resulting matrix, we will need to perform the appropriate multiplication and addition operations. The product of a row from the first matrix and a column from the second matrix becomes the corresponding element in the result matrix.

Check the Result

Once the multiplication is complete, examine the resulting matrix. Because \(B\) is the inverse of \(A\), the resulting matrix should be an identity matrix.

Key concepts

Matrix Multiplication

Matrix multiplication is a cornerstone concept in linear algebra that involves the combination of two matrices to produce a third matrix. Unlike ordinary multiplication, which simply scales a number, matrix multiplication combines two sets of numbers in a way that reflects the composition of linear transformations.

The process of multiplying matrices involves taking the rows of the first matrix and the columns of the second matrix and performing a series of multiplications and additions. For any element \( c_{ij} \) in the product matrix \( C \) that arises from multiplying matrix \( A \) and matrix \( B \) (where \( A \) has dimensions \( m \times n \) and \( B \) has dimensions \( n \times p \) ), it is calculated by summing up the products of the corresponding entries from the \( i \)th row of \( A \) and the \( j \)th column of \( B \) :
\[ c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \dots + a_{in}b_{nj} \]
So, when we multiply matrices \( A \) and \( B \) from the original problem, we are looking for a resulting matrix which should reflect the composition of the transformations represented by \( A \) and \( B \) if indeed \( B \) is the inverse of \( A \).

The ability to multiply matrices opens the door to solving systems of linear equations, understanding transformations in space, and much more. We must also remember that matrix multiplication is not commutative, meaning that in general, \( AB \) does not equal \( BA \). Hence, the order of multiplication is crucial.

Identity Matrix

The identity matrix is the matrix equivalent of the number 1 in regular multiplication. It's a special square matrix that, when multiplied by another compatible matrix, leaves the latter unchanged. It's denoted as \( I \) and is characterized by having ones on the diagonal and zeros elsewhere.

This concept is integral to the understanding of matrix inverses. If multiplying matrix \( A \) by matrix \( B \) yields the identity matrix (\( AB = I \)), then \( B \) is said to be the inverse of \( A \). In essence, multiplying by the inverse 'undoes' the transformation effected by the original matrix, which relates closely to the concept of reversal in mathematical operations.

The identity matrix serves a role similar to that of the number 1 in multiplication and division - it is the neutral element that leaves other elements unchanged when combined multiplicatively. If our calculated product in the original exercise is indeed the identity matrix, then it attests to the fact that the second matrix is the inverse of the first, fulfilling the exercise's requirement.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It's fundamental in areas such as science, engineering, and applied mathematics. The concepts of linear algebra extend to analytical geometry, computer science, economics, and many other disciplines, offering a means to solve complex problems through systemic and ordered approaches.

One of the key tools in linear algebra is the concept of a matrix inverse. The inverse matrix of a matrix \( A \) is another matrix, often denoted as \( A^{-1} \), such that when \( A \) is multiplied by \( A^{-1} \) the result is the identity matrix. This operation is significant because it allows us to solve matrix equations like \( AX = B \) by simply multiplying both sides by the inverse of \( A \) to find \( X \) : \( X = A^{-1}B \).

Finding the inverse of a matrix, like in the provided exercise, can be analogous to finding the reciprocal of a real number—just as we divide by a number's reciprocal to 'cancel it out,' we can multiply by a matrix's inverse to isolate a variable or 'undo' a transformation. The ability to manipulate and understand these relationships is central to mastering linear algebra. Plainly, linear algebra provides the language and tools for describing and solving a wide array of linear systems and geometric transformations.