Matrices and Determinants

In my recent two-day lesson, I introduced the concept of matrices and determinants, focusing on how to compute determinants for both 2×2 and 3×3 matrices. First, I presented a 2×2 matrix A: A = [ 1 2 ] [ 3 -4 ] To find the determinant of a 2×2 matrix, we use the formula: det(A) = (a × d) − (b × c) So for matrix A: det(A) = (1 × -4) − (2 × 3) det(A) = -4 − 6 det(A) = -10 Therefore, the determinant of matrix A is -10. Next, I introduced a 3×3 matrix B: B = [ 1 2 3 ] [ 0 3 1 ] [ 5 2 4 ] To find the determinant of a 3×3 matrix, we expand along the first row using cofactors: det(B) = 1 × | 3 1 | − 2 × | 0 1 | + 3 × | 0 3 | | 2 4 | | 5 4 | | 5 2 | Now compute each minor determinant: First minor: | 3 1 | = (3 × 4) − (1 × 2) = 12 − 2 = 10 | 2 4 | Second minor: | 0 1 | = (0 × 4) − (1 × 5) = 0 − 5 = -5 | 5 4 | Third minor: | 0 3 | = (0 × 2) − (3 × 5) = 0 − 15 = -15 | 5 2 | Now substitute back: det(B) = 1(10) − 2(-5) + 3(-15) det(B) = 10 + 10 − 45 det(B) = -25 Therefore, the determinant of matrix B is -25. In this lesson, students learned how determinants help us understand important properties of matrices, such as whether a matrix is invertible. A non-zero determinant means the matrix has an inverse, while a zero determinant means it does not. These examples provide a solid foundation for working with matrices in algebra and further mathematical studies.