Problem 3 - Discrete Mathematics with Applications

Question

Find the additive inverse of each matrix. $$\left[\begin{array}{rr} 2 & -3 \\ 0 & 4 \end{array}\right]$$

Step-by-Step Solution

Answer

The additive inverse of the matrix \(A = \left[\begin{array}{ll}{2} & {-3} \\\ {0} & {4}\end{array}\right]\) is \(B = \left[\begin{array}{ll}{-2} & {3} \\\ {0} & {-4}\end{array}\right]\).

Step 1: Identify the matrix dimensions

Find the dimensions of the given matrix A. In this case, A has dimensions 2x2, as there are 2 rows and 2 columns.

Step 2: Create a matrix filled with zeros of the same dimensions

Create a matrix of the same dimensions (2x2) filled with zeros, which is called the zero matrix. We'll call it Z. $$ Z = \left[\begin{array}{ll}{0} & {0} \\\ {0} & {0}\end{array}\right] $$

Step 3: Subtract matrix A from the zero matrix

Subtract each element of matrix A from the corresponding element of matrix Z. This will result in the additive inverse of A which we'll call B. $$ B = Z - A $$ So the matrix B is: $$ B = \left[\begin{array}{ll}{0-2} & {0-(-3)} \\\ {0-0} & {0-4}\end{array}\right] = \left[\begin{array}{ll}{-2} & {3} \\\ {0} & {-4}\end{array}\right] $$ So, the additive inverse of matrix A is given by the matrix B: $$ B = \left[\begin{array}{ll}{-2} & {3} \\\ {0} & {-4}\end{array}\right] $$

Key Concepts

Matrix Dimensions

Understanding the dimensions of a matrix is fundamental to working with matrices. The dimensions tell us the number of rows and columns in a matrix, described as 'm x n', where 'm' is the number of rows and 'n' is the number of columns. For example, if a textbook exercise presents
\[ A = \left[\begin{array}{cc} 2 & -3 \ 0 & 4 \end{array}\right] \],
this matrix A is a 2x2 matrix, meaning it has 2 rows and 2 columns. When identifying the additive inverse of a matrix, it's crucial to maintain the same dimensions to ensure a valid matrix operation. Therefore, the initial step in finding the additive inverse involves recognizing these dimensions.

Zero Matrix

A zero matrix acts as the identity element of addition for matrices. It consists entirely of zeros and will have the same dimensions as the matrix for which we seek the additive inverse. The zero matrix plays a pivotal role in finding the additive inverse because, when a matrix is subtracted from the zero matrix of the same dimensions, the resultant matrix will be the additive inverse of the original matrix. That is, if \[ Z \] is a zero matrix and \[ A \] is any matrix with the same dimensions, the additive inverse \[ B \] is found using the equation \[ B = Z - A \].
In our example, the zero matrix is \[ Z = \left[\begin{array}{cc} 0 & 0 \ 0 & 0 \end{array}\right] \], corresponding in size to matrix A.

Matrix Subtraction

Matrix subtraction involves subtracting corresponding elements of two matrices with the same dimensions. When you subtract matrix A from the zero matrix Z, element by element, the process is akin to changing the sign of each element in matrix A. This operation is what leads to finding the additive inverse of matrix A, now defined as matrix B.
Following the provided steps, matrix B, the additive inverse of A, is calculated by:
\[ B = \left[\begin{array}{cc} {0-2} & {0-( -3)} \ {0-0} & {0-4} \end{array}\right] \],
which simplifies to \[ B = \left[\begin{array}{cc} { -2} & { 3} \ { 0} & { -4} \end{array}\right] \].
The concept of subtraction here is crucial, as it helps students understand that obtaining the additive inverse of a matrix involves reversing the sign of all the elements within the matrix.