Method of Minor Matrices and Coefficients
This method requires the student to choose a row or column. It is best to choose the one which has the most entries of zero, if any, because those terms do not have to be computed. Take the non-zero entry in the row or column, multiply it by the appropriate sign, based on raising -1 to the power of the sum of row and column number. Then, multiply it by the determinant of the 2 x 2 minor matrix containing the entries outside of the chosen row and column. Repeat for any other non-zero entries in the chosen row or column. Finally, sum all of these products to obtain the determinant of the 3x3 matrix.
Method of Diagonals
A less complicated alternative is to use the method of diagonals. Although it would be nice to use this to compute determinants of matrices of higher order, it unfortunately only works for 3x3 square matrix. However, this shortcut method can be used to compute the minor 3x3 matrices derived from higher order matrices. It is a technique worth learning.
The first step is to rewrite the first two columns of entries to the right of the determinant matrix. Then, start with the upper left entry, in the first row and first column, and take the product of the three diagonal entries, moving downard from left to right. Repeat with the second and third entries in the first row, then sum the three products.
Next, start with the lower left entry and compute the three diagonal products, moving upward from left to right. Sum the three products, then subtract this sum from the sum of the previous sum.
Based on my experience as a private math tutor in Las Vegas since 2008, the method of diagonals is normally taught at the high school level. It is a less complicated method, but the tradeoff is that it doesnʻt work for higher order square matrices. The method of minor matrices and coefficients is typically taught at the high school honors and college level.
A third method for solving matrix determinants is Cramer's Rule, which I will not cover in this post.
The following is based on my private practice as a math tutor in Las Vegas since 2008, as well as teaching experience in the classroom. Addition and multiplication of monomials requires the novice algebra student to shift gears between adding and multiplying coefficients, as well as determining when addition of exponents is in order. This is demonstrated in the image above. In the first case (in black) the sum of monomials requires the coefficients, 7 and 3, to be added, but not the exponents, 2 and 2. In the second case (in blue), the product of monomials requirest the coefficients, 7 and 3, to be multiplied, and the exponents, 2 and 2, to be added. Since the second case has a combination of multiplication (of coefficents) and addition (of exponents), it's understandable why novice algebra students can get mixed up. Through the repetition of correctly solving enough of these operations over time, students develop the proficiency in knowing when to add or multiply coefficients, and when to add exponents.
Mr. Ken Marciel