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**Additional info for Linear algebra c-1 - Linear equations, matrices and determinants**

**Sample text**

2. (a) We shall solve the equation ⎛ ⎞ x1 1 2 4 3 ⎜ x2 AX = ⎝ 2 4 3 6 ⎠ ⎜ ⎝ x3 3 6 2 9 x4 ⎛ The total matrix ⎛ 1 2 4 ⎝ 2 4 3 3 6 2 ⎞ ⎛ ⎞ 7 ⎟ ⎟ = b1 = ⎝ 4 ⎠ . ⎠ 1 is equivalent to ⎞ ⎛ 7 ∼ 3 1 2 4 3 4 ⎠ R2 := 2R1 − R2 ⎝ 0 0 5 0 6 1 9 R3 := 3R1 − R3 0 0 10 0 ⎞ 7 10 ⎠ , 20 corresponding to the system x1 + 2x2 + 4x3 x3 + 3x4 = 7, = 2, thus x3 = 2 and x1 + 2x2 + 3x4 = −2. Choosing x2 = s and x4 = t, the set of solutions is {(−1 − 2s − 3t, s, 2, t) | s, t ∈ R}. (b) In this case ⎛ 1 ⎝ 2 3 the total matrix is equivalent to ⎞ ⎛ 7 ∼ 2 4 3 1 2 4 3 4 ⎠ R2 := 2R1 − R2 ⎝ 0 0 5 0 4 3 6 2 R3 := 3R1 − R3 6 2 9 0 0 10 0 ⎞ 7 10 ⎠ , 19 and the set of solutions is the empty set.

11 ⎛ 1 A=⎝ 4 7 2. Matrices Solve the matrix equation AX = 0, where ⎞ 2 3 5 6 ⎠ 8 9 and 0 is the (3 × 3) zero matrix. Find the rank of any solution matrix X. Since the zero matrix of coeﬃcients ⎛ ⎞ 1 2 3 ⎝ 4 5 6 ⎠ ⎛ 7 8 9 ⎞ 1 2 3 ⎝ 3 3 3 ⎠ ⎛ 3 3 3 ⎞ 1 2 3 ⎝ 1 1 1 ⎠ ⎛ 0 0 0 ⎞ 1 2 3 ⎝ 0 1 2 ⎠ 0 0 0 consists of three zero columns, we consider the equivalent matrix of the matrix ∼ R2 R3 ∼ R2 R3 := R2 − R1 := R3 − R2 := R2 /3 := R2 − R3 ∼ R := R1 − R2 ∼ R1 := R1 − R2 ⎛ ⎞ 1 1 1 ⎝ 0 1 2 ⎠. 0 0 0 Hence a particular column in the X matrix must necessarily satisfy x1 + x2 x2 + x3 + 2x3 = 0, = 0.

0 1 It follows that the unique solution is ⎛ ⎞ −1 −3 1 ⎠. com 55 Linear Algebra Examples c-1 2. 25 Given the matrices ⎛ ⎞ ⎛ ⎞ 1 2 −2 2 6 4 a ⎠, A=⎝ 1 B = ⎝ b 13 + a ⎠ , 0 −2 1 −2 −4 where a and b are arbitrary real numbers. 1. Find the pairs (a, b), for which the matrix equation AX = B has precisely one solution. 2. Find the complete solution of the matrix equation for (a, b) = (−3, 4). 1. The total matrix is equivalent to ⎞ ⎛ 2 6 1 2 −2 ∼ b 13 + a ⎠ 4 a (A | B) = ⎝ 1 R2 := R2 − R1 −2 ⎞−4 0 −2 1 ⎛ 2 6 1 2 −2 ∼ ⎝ 0 b−2 7+a ⎠ 2 a+2 R3 := R2 + R3 −2 −4 ⎞ 1 ⎛ 0 −2 2 6 1 2 −2 ⎝ 0 2 a+2 b−2 7+a ⎠ b−4 3+a 0 0 a+3 It follows that the matrix of coeﬃcients is of rank 3 for a = −3, hence there exists precisely one solution, when a = −3 and b ∈ R.

### Linear algebra c-1 - Linear equations, matrices and determinants by Mejlbro L.

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