You can break down any mathematical equation into terms, coefficients, variables and solutions. For example, the equation:
5x+3y=2
has terms "5x" and "3y," coefficients "5" and "3," the variables "x" and "y" and the solution is "2." A linear equation has a single variable in each term and no term squared, cubed or taken to any power. Linear equations have a single variable corresponding to each term of the equation, but the equation can have more than one term. When you have more than one linear equation, you have a system of linear equations.
Traditionally, you write out each linear equation one on top of the other. You can then write out the system as a matrix by separating the coefficients from the variables. Rewrite the equations, one on top of the other, but this time using only the coefficients of the equation, not including the solution. For example, the system of linear equations:
5x + 19y =3
3x +2y =2
has a corresponding matrix:
5 19
3 2
Mathematicians define each matrix by its columns and rows. The columns of a matrix include the numbers stacked on top of each other vertically while the rows include the numbers next to each other horizontally. In matrix form, the coefficients become "entries." You define each entry by the row and column it is in. For example, the first entry is in the first row and the first column, the entry to the right of it is in the first row and second column. Every entry has a corresponding row and column placement.
A diagonal matrix has numbers not equal to zero down the main diagonal. The main diagonal starts at the entry in the first column and the first row and includes all the entries from that corner to the bottom right corner. Every other entry in the matrix equals zero. All the zeroes in a diagonal matrix make it easy to mathematically manipulate. For example, adding two diagonal matrices together produces another diagonal matrix. To add two diagonal matrices together, add corresponding entries of each one's diagonal and write the solution in the same entry spot of the solution matrix.
Triangular matrices share properties with diagonal matrices, but only half of the matrix has zeroes. Mathematicians subdivide triangular matrices into two categories -- upper triangular and lower triangular. In an upper triangular matrix, every entry below the main diagonal equals zero, and in a lower triangular matrix, every entry above the main diagonal equals zero. Like diagonal matrices, adding together two triangular matrices produces a triangular matrix.
You can solve a system of linear equations by performing addition, subtraction, multiplication and division on the entries of the matrix. The end result produces a diagonal matrix with only ones down the main diagonal. Linear algebra books call this "the identity matrix." Each one on the main diagonal will have a corresponding entry in the solutions column, and those numbers are the solutions to the original system of linear equations.