In its most basic form, a matrix consists of rows and columns of numbers often referred to as an array of numbers. To visualize, imagine the arrangement of a Sudoku puzzle. If you arrange the numbers one on top of the other in vertical lines and set each line side by side you say the numbers are in columns. If you prefer to write them horizontally next to one and other you call each horizontal line of numbers a row of the matrix. Either way, the end result will resemble the Sudoku puzzle.
The coefficients of an equation are the numerical part. For example, in the equation:
3x^2 +5x, three and five are the coefficients. If you have more than one equation, then you have a system of equations. Early mathematicians tried to solve systems of equations by combining their coefficients into rectangular arrays of numbers; this led to the first matrices.
Mathematical matrices have roots in the math of the Babylonians of four hundred B.C. However, most of the credit toward their inception goes to the Han Dynasty of ancient China. By studying the coefficients of systems of equations, these ancient mathematicians came up with a simpler way to find solutions to a system of equations. By placing the coefficients into an array of rows and columns, they could solve several equations at the same time. The term matrix did not appear until 1850 from the mathematician James Joseph Sylvester. Although they did not call them matrices, mathematicians still followed the same principle.
The initial invention of these rectangular arrays of numbers allowed mathematicians to more easily handle multiple equations. Eventually, mathematicians further developed the applications and theories of matrix algebra to deal with a large range of math problems. For example, Frederick Gauss developed a system of matrix mathematics to deal with astronomy problems. Other early uses included studying the effects of wind on airplanes and advanced calculus problems.