Definition. A matrix is upper triangular if everything below the main diagonal is 0. (Entries on and above the main diagonal may or may not be zeroes).
Similarly, a matrix is lower triangular if everything above the main diagonal is 0. (Entries on and below the main diagonal may or may not be zeroes).
Let A be a triangular matrix.
detA=a11a22...ann=i=1∏naii. This is the same computation used to find the determinant of a diagonal matrix.
Theorem. Let A be a square matrix.
- If a multiple of one row of A is added to another row to form a matrix B, then detB=detA.
- If two rows of A are interchanged to produce B, then detB=−detA.
- If one row of A is multiplied by k to produce B, then detB=k detA
By row-reducing a matrix into an upper triangular matrix and keeping these rules in mind, we can find its determinant much more quickly.
Ex.
AdetA=⎝⎛1−2−1−487−290⎠⎞=∣∣∣∣∣∣1−2−1−487−290∣∣∣∣∣∣=∣∣∣∣∣∣10−1−407−250∣∣∣∣∣∣=−∣∣∣∣∣∣1−10−470−205∣∣∣∣∣∣=−∣∣∣∣∣∣100−430−2−25∣∣∣∣∣∣=−(1)(3)(5)=−15.
BdetB=⎝⎛1−1−2−478−209⎠⎞=∣∣∣∣∣∣1−1−2−478−209∣∣∣∣∣∣=−∣∣∣∣∣∣1−2−1−487−290∣∣∣∣∣∣=−detA=−(−15)=15.
Theorem. Let A be a square matrix.
- A is invertible if and only if detA=0.
- detAT=detA.
- detAB=detA detB.
ABAB=(cosθsinθ−sinθcosθ)=(cosψsinψ−sinψcosψ)=(cosθcosψ−sinθsinψsinθcosψ+cosθsinψ−cosθsinψ−sinθcosψ−sinθsinψ+cosθcosψ)=(cos(θ+ψ)sin(θ+ψ)−sin(θ+ψ)cos(θ+ψ)) This result maxes sense because multiplying the matrices is essentially combining the two rotations, which produces a new rotation matrix of the two angles added together.
detAdetBdetAB=cos2θ+sin2θ=1=cos2ψ+sin2ψ=1=detA detB=1