Matrices

• A square matrix is a matrix where the number of rows and columns are the same
• A zero matrix is a matrix where all its numbers are zero and it is denoted by 0
• An identity matrix is a square matrix in which the numbers in the diagonal starting top left are 1 and the rest are 0, denoted by I
• To add or subtract matrices, you add or subtract the corresponding elements in each, you can only add and subtract matrices the same size
• To multiply a matrix by a scalar, you multiply every element in the matrix by that scalar
• Matrices can be multiplied if the number of columns in the first matrix is the same as the number of rows in the second matrix
• To find the product of two matrices, you multiply the elements in each row of the first matrix by the corresponding elements in each column of the second matrix and then adding the results together
• For the 2x2 matrix M the determinant is ad-bc

  
• If the determinant of M is equal to zero, M is a singular matrix
• If the determinant of M does not equal zero, the matrix is non-singular
• You find the determinant of a 3x3 matrix by reducing the 3x3 determinant to 2x2 determinants using the formula:

  
• The minor of an element in a 3x3 matrix is the determinant of the 2x2 matrix that remains after the row and column containing that element have been crossed out
• The inverse of a matrix M is:

  
• If A and B are non-singular matrices, then:

  
• The transpose of a matrix is found by interchanging the rows and columns
• The first step of finding the inverse of 3x3 matrix A is:

  • Find the determinant of A, detA
• The second step of finding the inverse of 3x3 matrix A is:

  • Form the matrix of minors of A, M
  • In forming the matrix of minors of A, you replace each of the nine elements of A with its minor
• The third step of finding the inverse of a 3x3 matrix A is:

  • From the matrix of minors form the matrix of cofactors, C, by changing the signs of some of the elements according to the rule of alternating signs:

  
• The fourth step of finding the inverse of a 3x3 matrix A is:

  • Write down the transpose of the matrix of cofactors
• The final step of finding the inverse of a 3x3 matrix A is:

  • The inverse of the matrix A is given by the formula: