If #A# is a matrix, where #A=[(a_11,a_12,a_13), (a_21,a_22,a_23), (a_31,a_32,a_33)]#, then we define the determinant of the matrix to be
#|A|=|(a_11,a_12,a_13), (a_21,a_22,a_23), (a_31,a_32,a_33)|#
#implies |A|=a_11|(a_22,a_23), (a_32,a_33)|-a_12|(a_21,a_23), (a_31,a_33)|+a_13|(a_21,a_22), (a_31,a_32)|#
#implies |A|=a_11(a_22a_33-a_32a_23)-a_12(a_21a_33-a_31a_23)+a_13(a_21a_32-a_31a_22)#
And then simply this expression.
Now, let #A=[(-5,4,1), (4,7,0), (-3,4,-1)]#
#implies |A|=|(-5,4,1), (4,7,0), (-3,4,-1)|=-5|(7,0), (4,-1)|-4|(4,0), (-3,-1)|+1|(4,7), (-3,4)|#
#implies |A|=-5{7*(-1)-4*0}-4{4(-1)-(-3)*0}+1{4*4-(-3)*7}#
#implies |A|=-5(-7-0)-4(-4-0)+(16+21)#
#implies |A|=35+16+37=88#
Hence, determinant of the given matrix is #88.#
