For each of the following 3 × 3 matrices, find the determinant and indicate the rank of the matrix.

Setting down the original matrix and to the right of it repeating the first two columns,

Then proceeding as in Example 2, with Fig. 6-3,

With |A| ≠ 0, A is nonsingular. All three rows and columns are linearly independent and ρ(A) = 3.

Setting up the extended matrix in Fig. 6-4 and calculating,

B is nonsingular; ρ(B) = 3.

Setting up the extended matrix in Fig. 6-5 and calculating,

With |C| = 0, C is singular and all three rows and columns are not linearly independent. Hence ρ(C)
≠ 3. Though the determinant test points to the existence of linear dependence, it does not specify the nature of the dependency. Here row 3 is 1.75 times row 2. To test if any two rows or columns in C are independent, apply the determinant test to the various submatrices. Starting with the 2 × 2 submatrix in the upper left-hand corner,

With |C1| ≠ 0, there are two linearly independent rows and columns in C and ρ(C) = 2.

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(a) 75 A = 52 1 4 49 8 6 3