If A⁻¹ = [[1, 2], [3, 4]] and B⁻¹ = [[−2, 4], [6, 8]], find (a) (A^T)⁻¹, (b) (AB)⁻¹, (c) (2B)⁻¹. If P = [[3…

Question

If A⁻¹ = [[1, 2], [3, 4]] and B⁻¹ = [[−2, 4], [6, 8]], find (a) (A^T)⁻¹, (b) (AB)⁻¹, (c) (2B)⁻¹. If P = [[3, −2], [0, 2]], find P⁻¹.

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the matrix problems. **1) Given A⁻¹ = 1 & 2; 3 & 4 and B⁻¹ = −2 & 4 6 & 8 .** a) Find (A^T)⁻¹ Step 1: Use the property (A^T)⁻¹ = (A⁻¹)^T. (A^T)⁻¹ = ( 1 & 2 3 & 4 )^T Step 2: Transpose the matrix A⁻¹ by swapping its rows and columns. (A^T)⁻¹ = 1 & 3 2 & 4 The result is 1 & 3 2 & 4 . b) Find (AB)⁻¹ Step 1: Use the property (AB)⁻¹ = B⁻¹A⁻¹. (AB)⁻¹ = −2 & 4 6 & 8 1 & 2 3 & 4 Step 2: Perform the matrix multiplication. (AB)⁻¹ = (−2)(1) + (4)(3) & (−2)(2) + (4)(4) (6)(1) + (8)(3) & (6)(2) + (8)(4) (AB)⁻¹ = −2 + 12 & −4 + 16 6 + 24 & 12 + 32 (AB)⁻¹ = 10 & 12 30 & 44 The result is 10 & 12 30 & 44 . c) Find (2B)⁻¹ Step 1: Use the property (kB)⁻¹ = 1/k B⁻¹. Here k=2. (2B)⁻¹ = 1/2 B⁻¹; (2B)⁻¹ = 1/2 −2 & 4 6 & 8 Step 2: Multiply each element of B⁻¹ by 1/2. (2B)⁻¹ = (−2)/(2) & 4/2 6/2 & 8/2 (2B)⁻¹ = −1 & 2 3 & 4 The result is −1 & 2 3 & 4 . **2) If P = 3 & −2 0 & 2 , find P⁻¹.** Step 1: Calculate the determinant of P. For a 2 × 2 matrix a & b c & d , the determinant is ad − bc. det(P) = (3)(2) − (−2)(0) = 6 − 0 = 6; Step 2: Use the formula for the inverse of a 2 × 2 matrix: P⁻¹ = (1)/(det(P)) d & −b −c & a . P⁻¹ = 1/6 2 & −(−2) −0 & 3 P⁻¹ = 1/6 2 & 2 0 & 3 Step 3: Multiply each element by 1/6. P⁻¹ = 2/6 & 2/6 0/6 & 3/6 P⁻¹ = 1/3 & 1/3 0 & 1/2 The inverse of P is 1/3 & 1/3 0 & 1/2 .

If A⁻¹ = [[1, 2], [3, 4]] and B⁻¹ = [[-2, 4], [6, 8]], find (a) (A^T)⁻¹, (b) (AB)⁻¹, (c) (2B)⁻¹. If P = [[3, -2], [0, 2]], find P⁻¹.

Answers

Step-by-step solution

Part a· 2 steps

Use the property $(A^T)^{-1} = (A^{-1})^T$ .

Transpose the matrix $A^{-1}$ by swapping its rows and columns.

Part b· 2 steps
$\begin{pmatrix} 10 & 12 \\ 30 & 44 \end{pmatrix}$

Use the property $(AB)^{-1} = B^{-1}A^{-1}$ .

Perform the matrix multiplication.

Part c· 5 steps
$\begin{pmatrix} -1 & 2 \\ 3 & 4 \end{pmatrix}$

Use the property $(kB)^{-1} = \frac{1}{k} B^{-1}$ .

Multiply each element of $B^{-1}$ by $\frac{1}{2}$ .

Calculate the determinant of $P$ .

Use the formula for the inverse of a $2 \times 2$ matrix

Multiply each element by $\frac{1}{6}$ .

Try one yourself

Still stuck on this one?

More Mathematics Questions

If A⁻¹ = [[1, 2], [3, 4]] and B⁻¹ = [[-2, 4], [6, 8]], find (a) (A^T)⁻¹, (b) (AB)⁻¹, (c) (2B)⁻¹. If P = [[3, -2], [0, 2]], find P⁻¹.

Answers

Step-by-step solution

Part a· 2 steps

Use the property (AT)−1=(A−1)T(A^T)^{-1} = (A^{-1})^T(AT)−1=(A−1)T.

Transpose the matrix A−1A^{-1}A−1 by swapping its rows and columns.

Part b· 2 steps(10123044)\begin{pmatrix} 10 & 12 \\ 30 & 44 \end{pmatrix}(1030​1244​)

Use the property (AB)−1=B−1A−1(AB)^{-1} = B^{-1}A^{-1}(AB)−1=B−1A−1.