Matrix Equations

Matrices Equations Revision

Theory and Concepts

Matrices are an array of numbers or expressions arranged in rows and columns. They are used to represent and handle linear equations and transformations.
A matrix equation represents a system of linear equations.

Matrix Addition and Subtraction

Two matrices can be added or subtracted if and only if their orders (number of rows and columns) are exactly the same.
For addition, corresponding elements in each of the matrices are added together.
In subtraction, the corresponding elements are subtracted.

Matrix Multiplication

Multiplication of two matrices is possible if the number of columns in the first matrix is equal to the number of rows in the second matrix.
Multiply corresponding elements then sum them to find the resultant matrix element.

Identity Matrix

An identity matrix is a square matrix in which all the elements of the main diagonal are ones and all other elements are zeros.
When a matrix is multiplied by an identity matrix, the original matrix remains the same.

Inverse Matrices

The inverse of a matrix A, denoted as A^-1, is the matrix such that when it is multiplied by A, it results in the identity matrix.
Not all matrices have an inverse. A matrix has an inverse only if its determinant is non-zero.

Solving Matrix Equations

To solve a matrix equation of the form AX = B, where A is a matrix, B is a matrix, and X is the matrix of unknowns, you can use the inverse of A.
Multiply both sides of the equation by A^-1 to isolate X: X = A^-1B is the solution of the matrix equation AX = B.

Determinant of a Matrix

The determinant is a special number that can be calculated from a square matrix.
It is used to find the inverse of a matrix and to solve systems of equations.

Applications of Matrices

Matrices are widely used in geometry, physics, and computer graphics applications. It’s also utilized in calculating the battery power outputs and in representing the real-world data’s like the population of people, etc.

Remember to practise as many matrix calculations and manipulations as possible to get comfortable with matrices. Make sure you understand how to multiply matrices, how to find the inverse of a matrix, and how to use these operations to solve matrix equations.