Three facts make the determinant useful. It does not change when a matrix is transposed, it turns a product of matrices into a product of numbers, and it is never zero for a matrix that can be undone. Each has a picture: area factors multiply when maps are composed.

Enlarging a photograph by 2 and then by 3 enlarges it by 6 in all. The determinant of a product is the product of the determinants for the same reason: it is the enlargement factor of the map.

Both rules hold for all $n\times n$ matrices. The transpose rule holds because transposing turns each term of the definition into a term with the same entries and the same sign. It means that everything said about rows is true of columns, so column operations may be used exactly like row operations. The product rule is the multiplication of area factors when two maps are composed. It is stated here and used; it is proved later in the course, once every invertible matrix has been written as a product of row operations.

• $\det(A^k)=(\det A)^k$ for every power $k$.
• $$, even though and are usually different.

Apply the product rule to a matrix times its inverse. The product is the identity, whose determinant is 1, so the two determinants multiply to 1, and neither of them can be zero.

This is what it means for $A^{-1}$ to be the inverse of $A$.

Take the determinant of both sides; the identity is diagonal with ones on the diagonal.

The product rule turns the determinant of a product into a product of determinants.

Two numbers whose product is $1$ are both non-zero, and each is the reciprocal of the other.

Transposing a matrix keeps its determinant. So everything the last lesson said about rows holds for columns: exchanging two columns flips the sign, scaling a column scales it, and adding a multiple of one column to another keeps it.

The determinant of a product is the product of the determinants. Geometrically, doing one map after another multiplies the two area factors.

• $\det A^{T}=\det A$, so column operations obey the same rules as row operations.
• $\det(AB)=\det A\,\det B$, and so and .

Great work! You now know how to:

• use $\det A^{T}=\det A$ and work with columns as with rows;
• use $\det(AB)=\det A\det B$ and its consequences for powers and inverses;