Problem 78 - Precalculus with Limits

Question

TRUE OR FALSE? In Exercises 77 and 78, determine whether the statement is true or false. Justify your answer. If you multiply two square matrices and obtain the identity matrix, you can assume that the matrices are inverses of one another.

Step-by-Step Solution

Answer

The statement is True

Step 1: Defining an Identity Matrix

An identity matrix is a square matrix in which all the elements of the principal (main) diagonal are ones and all other elements are zeros. It is denoted as \(I_n\) or \(I\), where \(n\) denotes the order of the square matrix.

Step 2: Understanding Matrix Inversion

Two n × n matrices \(A\) and \(B\) are said to be inverse of each other if their multiplication results in an identity matrix of order \(n\). That is, if \(AB=BA=I_n\), then matrix \(B\) is an inverse of matrix \(A\) and vice versa.

Step 3: Apply this information on given statement

Given the statement in the question and our definitions from above, it can be confirmed that if two square matrices, when multiplied, yield the identity matrix, then they are indeed inverses of each other.

Key Concepts

Understanding the Identity Matrix

The identity matrix plays a pivotal role in the world of linear algebra. It is a special kind of square matrix where all the elements on the main diagonal are '1's and all other elements are '0's. This structure is critical because multiplying any matrix by the identity matrix results in the original matrix. Simply, it’s equivalent to the number '1' in multiplication for real numbers. For instance, an identity matrix of size 3×3 looks like this:
\[I_{3} = \begin{pmatrix}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1d{pmatrix}\]
When we multiply any matrix, say matrix A, by the identity matrix, we get the same matrix A as a result, which can be expressed as \(AI = IA = A\). This property is essential when considering inverse matrices because the inverse of a matrix essentially 'undoes' the effect of the original matrix, resulting in the identity matrix.

Decoding Matrix Multiplication

Matrix multiplication is not as straightforward as multiplying numbers. It involves a systematic process where the elements of the rows of one matrix are multiplied with corresponding elements of the columns of another and then summed up to create the elements of the resultant matrix. Important to remember is that the order of multiplication matters in matrix multiplication; in most cases, \(AB\) does not equal \(BA\).
Here's a simplified explanation:

• Select the elements of a row from the first matrix.
• Then, select the corresponding elements of a column from the second matrix.
• Multiply these selected elements pair-wise and sum up all the products.
• The result of these operations gives you the element of the new matrix that corresponds to the position of the row and column you chose initially.

If you manage to get an identity matrix after multiplying two matrices \(A\) and \(B\), then mathematically, those matrices are considered inverses of each other since \(AB = I\) and \(BA = I\), where \(I\) is the identity matrix.

Characteristic of Square Matrices

Square matrices have the same number of columns and rows, leading to a 'square' format, and are a prerequisite for many key calculations in matrix algebra, including the calculation of matrix inverses. Inverses can only be found for square matrices, and not every square matrix has an inverse (these are known as 'singular' or 'non-invertible' matrices). A matrix that has an inverse is known as 'non-singular' or 'invertible'.
For example, a matrix \(A\) is a square matrix of order 2 if it has the form:
\[A = \begin{pmatrix}a_{11} & a_{12} \a_{21} & a_{22}d{pmatrix}\]
It has 2 rows and 2 columns. The characteristic property for identification of an invertible matrix is that when it multiplies with its inverse, the product is the identity matrix, fulfilling the equation \(AA^{-1} = A^{-1}A = I\), where \(I\) is the identity matrix of the same order.