System of Equations
The augmented matrix is already in row echelon form. Use back substitution to find (x,y,z).
[ 1 2 -1 | 2 ]
[ 0 1 3 | 7 ]
[ 0 0 2 | 4 ]
(2, 1, 2)
Add the matrices.
[ 3 2 ] [ 4 -1 ]
[ -1 5 ] + [ 2 3 ]
[ 7 1 ]
[ 1 8 ]
Use a determinant to determine whether the points are collinear.
(1, 1), (2, 2), (3, 3)
Collinear
Find the determinant.
[ 4 3 ]
[ -2 5 ]
26
Find the dimensions of the matrix.
[ 3 -2 5 ]
[ 1 7 0 ]
2 × 3
Use Cramer's Rule to solve:
2x + y = 7
3x - 2y = 0
(2, 3)
Subtract the matrices.
[ 8 3 ] [ 2 5 ]
[ -1 6 ] - [ -4 2 ]
[ 6 -2 ]
[ 3 4 ]
Use a determinant to determine whether the points are collinear.
(-1, 2), (2, 4), (5, 7)
Not collinear
Use the downs-minus-ups method to find the determinant.
[ 2 1 3 ]
[ 0 4 -1 ]
[ 2 5 1 ]
-8
Determine the value of each variable.
[ x + 2 y - 1 ] [ 8 4 ]
[ 5 3 ] = [ 5 k ]
x = 6, y = 5, k = 3
Use Cramer's Rule to solve:
x + y + z = 6
2x - y + z = 3
x + 2y - z = 2
(1, 2, 3)
Multiply the matrices.
[ 2 1 ] [ 3 0 ]
[ 4 -1 ] × [ 2 5 ]
[ 8 5 ]
[ 10 -5 ]
Use a determinant to find the area of the triangle with vertices:
(0, 0), (4, 0), (0, 6)
Area = 1/2 |24| = 12
Use the minor method to find the determinant.
[ 2 1 -1 ]
[ 3 0 2 ]
[ 1 4 1 ]
-29
Determine the value of each variable.
[ 2x - 1 y + 4 ] [ 11 9 ]
[ k + 2 3y ] = [ 8 15 ]
x = 6, y = 5, k = 6
Use row operations to put the augmented matrix into row echelon form. Then solve the system.
[ 2 3 | 13 ]
[ 4 -1 | 5 ]
(2,3)
Multiply the matrices.
[ 1 2 -1 ] [ 2 1 ]
[ 3 0 2 ] × [ -1 3 ]
[ 4 2 ]
[ -4 5 ]
[ 14 7 ]
Use a determinant to find the area of the triangle with vertices:
(-2, 1), (3, 4), (5, -2)
Area = 1/2 |-36| = 18
The determinant of the matrix is 23. Find x.
[ x 3 ]
[ 4 5 ] = 23
x = 7
Find A⁻¹ (the inverse of A) if it exists.
A = [ 3 1 ]
[ 2 1 ]
A⁻¹ = [ 1 -1 ]
[ -2 3 ]
Use row operations to put the augmented matrix into row echelon form. Then solve the system.
[ 1 1 1 | 6 ]
[ 2 3 1 | 11 ]
[ 3 2 2 | 11 ]
(-1, 3, 4)
Find AB + C.
A = [ 1 2 ] B = [ 3 -1 ] C = [ 2 1 ]
[ 0 -1 ] [ 2 4 ] [ 3 0 ]
[ 9 8 ]
[ 1 -4 ]
A rectangle has vertices:
A(1, 1), B(5, 3), C(4, 5), D(0, 3)
Use determinants to find the area of the rectangle.
Total Area = 5 + 5 = 10 square units
The determinant of the matrix is 21. Find x.
[ x 1 0 ]
det [ 2 3 1 ] = 21
[ 1 0 2 ]
x = 4
Find A⁻¹ (the inverse of A) if it exists.
A = [ -4 3 ]
[ 2 -5 ]
A⁻¹ = [ -5/14 -3/14 ]
[ -1/7 -2/7 ]