Properties of the Determinant

• if A is transformed to R through the used of row operations, then det(A) is related to det(R)

• Scaling a row

  • if B is obtained from A by multiplying a row by r, then det(B)=rdet(A)

    • (basically we can just reverse the scaling of matrices with determinant (determinant is proportional to the matrix ))
    • if its like a 4x4 matrix with det(2A) we must relise that 2A multiplies all entries by 2 $\therefore 2^4\det (A)$ .

• Row multiples

  • If two rows of a matrix are scalar multiplse of another, then $\det =0$

• REF

  • We can get the Determinant by reducing it to REF:

    • reduce a matrix (upper triangular)
      • then we can just get the product of the diagonal entries to get determinant

    • We have to keep in mind the operations done to the matrix and the what the reverse is so we can do them bac to the original

      • (follow the row operations bacwards to get Det)

Examples

• Using Upper triangular and the Properties of the determinant

  • Find the determinant of

$$\left[\begin{array}{ccc}1& 1& 1\\ 0& 0& 2\\ 0& 3& 4\end{array}\right]⟹\left[\begin{array}{ccc}1& 1& 1\\ 0& 3& 4\\ 0& 0& 2\end{array}\right]$$

• Now we can see that the matrix is uper triangular so the determinent is the products of the diagonal numbers
  • $\det =1\ast 3\ast 2=6$

• Using REF to find 4x4 det

  • find the determinant of

$$\begin{array}{r}\left[\begin{array}{cccc}0& 1& 2& 2\\ 2& 2& 2& 2\\ 1& 2& 5& 5\\ 2& 2& 2& 1\end{array}\right]⟹RREF\left[\begin{array}{cccc}1& 1& 1& 1\\ 0& 1& 2& 2\\ 0& 0& 2& 2\\ 0& 0& 0& -1\end{array}\right]\\ Det(REF)=1\ast 1\ast 2\ast -1=-2\\ Det(REF)\ast -1\ast 2=Det(A)\\ Det(A)=4\end{array}$$