3.7 – Evaluate Determinants and
Apply Cramer’s Rule
Associated with each square (n x n) matrix is
a real number called its determinant. The
determinant of a matrix A is denoted by det
A or by |A|.
The determinant of a 2 x 2 matrix is the
difference of the products of the elements of
the diagonals.
3.7 – Evaluate Determinants and
Apply Cramer’s Rule
3.7 – Evaluate Determinants and
Apply Cramer’s Rule
Example 1:
Evaluate the determinant of the matrix.
3.7 – Evaluate Determinants and
Apply Cramer’s Rule
Example 2:
Evaluate the determinant of the matrix.
3.7 – Evaluate Determinants and
Apply Cramer’s Rule
You can use a determinant to find the area of
a triangle whose vertices are points in a
coordinate plane.
3.7 – Evaluate Determinants and
Apply Cramer’s Rule
Example 3:
Off the coast of California lies a triangular
region of the Pacific Ocean where huge
populations of sea lions and seals live. The
triangle is formed by imaginary lines
connecting Bodega Bay, the Farallon
Islands, and Ano Nuevo Island, as shown
(next slide). Use the determinant to estimate
the area of the region.
3.7 – Evaluate Determinants and
Apply Cramer’s Rule
Example 3:
Units measured in miles.
3.7 – Evaluate Determinants and
Apply Cramer’s Rule
You can use determinants to solve a system
of linear equations. The method, called
called Cramer’s rule and named after the
Swiss mathematician Gabriel Cramer, uses
the coefficient matrix of the linear system.
3.7 – Evaluate Determinants and
Apply Cramer’s Rule
3.7 – Evaluate Determinants and
Apply Cramer’s Rule
Example 4:
Use Cramer’s rule to solve the system:
3.7 – Evaluate Determinants and
Apply Cramer’s Rule
3.7 – Evaluate Determinants and
Apply Cramer’s Rule
Example 5:
Use Cramer’s rule to solve the linear system.