What is the correct way to perform matrix multiplication?
To multiply two matrices, take the rows of the first matrix and columns of the second, multiply their corresponding entries and then sum those products. The resultant matrix's element in the i-th row and j-th column is obtained by multiplying the elements of the i-th row of the first matrix with the respective elements of the j-th column of the second matrix.
What are the basic operations one can perform on matrices?
Basic matrix operations include addition, subtraction, multiplication, division (more accurately, multiplying by the inverse of another matrix), and finding the determinant. One can also perform scalar multiplication and compute the transpose of a matrix.
What are the steps for computing the determinant of a matrix?
To compute the determinant of a matrix, firstly, ensure it's square (equal rows and columns). If it's a 2x2 matrix, multiply the diagonals and subtract. For matrices larger than 2x2, apply row reduction to upper triangular form, then multiply the diagonal elements. Alternatively, use expansion by minors for any size.
How do you find the inverse of a matrix?
To find the inverse of a matrix, first calculate its determinant. If the determinant is not zero, then compute the matrix of cofactors, transpose it to get the adjugate matrix, and finally divide each element of the adjugate matrix by the determinant.
How does one carry out the addition or subtraction of matrices?
To add or subtract matrices, simply add or subtract corresponding elements from each matrix. Ensure both matrices have the same dimensions. For example, if \(A\) and \(B\) are matrices, then \(C = A + B\) (or \(C = A - B\) for subtraction) where \(C_{ij} = A_{ij} + B_{ij}\) (or \(C_{ij} = A_{ij} - B_{ij}\)).