Answer: The square matrix you described, which contains ones along the main diagonal and zeros everywhere else, is called an identity matrix. It is denoted by the symbol "I" or sometimes "I_n" to indicate its size, where "n" represents the number of rows (which is also equal to the number of columns).
The purpose of an identity matrix in matrix algebra is similar to the purpose of the number 1 in arithmetic operations. It acts as the multiplicative identity for matrices, meaning that when you multiply any matrix by an identity matrix of the appropriate size, the resulting matrix is equal to the original matrix. In other words, it does not change the original matrix.
Explanation:
For example, let's consider a 3x3 identity matrix:
1 0 0
0 1 0
0 0 1
If we multiply this identity matrix by any 3x3 matrix, say:
2 4 6
1 3 5
0 1 2
The result will be the same matrix:
(1*2 + 0*1 + 0*0) (1*4 + 0*3 + 0*1) (1*6 + 0*5 + 0*2)
(0*2 + 1*1 + 0*0) (0*4 + 1*3 + 0*1) (0*6 + 1*5 + 0*2)
(0*2 + 0*1 + 1*0) (0*4 + 0*3 + 1*1) (0*6 + 0*5 + 1*2)
which simplifies to:
2 4 6
1 3 5
0 1 2
This property of the identity matrix is useful in many matrix operations, such as solving systems of linear equations, finding inverses of matrices, and performing matrix transformations. It serves as a fundamental tool in matrix algebra by preserving the original matrix properties while allowing for mathematical operations to be carried out.