Matrix arithmetic

• A matrix has dimensions [no of rows] X [no of columns]
• The determinant of a 2x2 matrix A is found using: det(A) = ad-bc
• The inverse of a 2x2 matrix is equal to the matrix with the lead diagonal transposed and the negative of each element on the reverse diagonal multiplied by 1 over the determinant if the determinant is not 0
• Scalar multiplication with a matrix is carried out by multiplying each element in the matrix by the scalar
• Multiplying two matrices together can only be done if the number of columns in the first matrix is equal to the number of rows in the second matrix
• To multiply two matrices, you multiply the first element in the current row of the first matrix by the corresponding element in the current column of the second matrix, and add the results
• Addition and subtraction of matrices is applied to each individual element
• To divide by a matrix, you multiply by its inverse
• A square matrix has the same number of rows and columns
• The identity matrix has 1s on the lead diagonal, and the rest of the elements are 0. It has no effect on a matrix when it is multiplied by it
• The zero matrix contains only 0s, and is the identity in matrix addition and subtraction
• Transposing a matrix flips the matrix along its lead diagonal
• det(AB) = det(BA) = det(A) X det(B)
• det(kA) = k^2 det(A)
• A matrix multiplied by its inverse gives the identity matrix
• To find the determinant of a 3X3 matrix, you take the top left element, delete the row and column that it appears in, find the determinant of the remaining 2X2 matrix and multiply it by the top left element of the original matrix. You repeat this with the top middle element and subtract the result, and then add the result from the top right element
• To find the inverse of a 3X3 matrix, you delete the row and column of each element and find the determinant of the remaining 2X2 matrix. You then change the sign of each element according to the alternating sign matrix. Transpose the matrix and multiply it by 1 over the determinant