3x3 Matrix Inverse Explained with Simple Animations

Welcome to understanding matrix determinants with Spark.E!A determinant is a special number we can calculate for any square matrix. Let's look at a three by three matrix A.For a three by three matrix, the determinant formula uses a method called cofactor expansion along the first row.The formula has three main terms, each involving a first-row element multiplied by a special two by two determinant.Let's work through an example with actual numbers.For the first term, we take a one one, which is 2, and multiply it by the determinant of the two by two matrix formed by removing row 1 and column 1.For the second term, we take negative a one two, which is negative 1, times its corresponding two by two determinant.The third term involves a one three, which is 3, times its two by two determinant.Adding all three terms together, we get a determinant of twenty-five.Since our determinant is not zero, this tells us that our matrix is invertible, meaning we can find its inverse.Now that we understand how to calculate the determinant, we're ready to move on to finding the adjugate matrix.To find the adjugate matrix, we first need to calculate the cofactors for each element.The sign of each cofactor follows a checkerboard pattern, alternating between positive and negative.Let's calculate the first cofactor. For position one-one, we take the determinant of the two-by-two matrix formed by removing row one and column one.After calculating all nine cofactors, we arrange them into the cofactor matrix.To create the adjugate matrix, we transpose the cofactor matrix, meaning we swap rows and columns.The resulting matrix is the adjugate matrix, which we'll use in the next step to find the inverse.Now that we have our determinant and adjugate matrix, we can compute the inverse.Recall that our determinant is 17And we calculated the adjugate matrix in the previous stepWe multiply the adjugate matrix by one over the determinantThis gives us our inverse matrix, with decimal approximations for clarityLet's verify our result by multiplying the original matrix by its inverseLet's look at how we get the first element of our identity matrixWe multiply each row of A by each column of A inverseWhen we complete all the multiplications, we get the identity matrix, with ones on the diagonal and zeros elsewhereThis verifies that we have found the correct inverse matrix