Which of the following matrices is invertible?

Question

BlackTom AI · Accepted Answer

Question: Which of the following matrices is invertible?

Option 1: [1,0][0,1]
- This is the identity matrix I2. Its determinant is 1*1 - 0*0 = 1, which is nonzero. A nonzero determinant implies the matrix is invertible. Therefore this option is consistent with invertibility.

Option 2: [1, 2] [2, 4]
- Interpreted as the matrix [[1, 2], [2, 4]]. The determinant is 1*4 - 2*2 = 4 - 4 = 0. A zero determinant means the matrix is singular and not invertible. This option is not invertible due to linear dependence between rows/columns.

Option 3: [0,0][0,0]
- This is the zero matrix. Its determinant is 0, and it maps every vector to the zero vector. A determinant of zero confirms non-invertibility. This option is clearly not invertible.

Option 4: [1,1][1,1]
- The matrix [[1, 1], [1, 1]] has determinant 1*1 - 1*1 = 0. The rows are identical, indicating redundancy and a lack of full rank. Hence it is not invertible.

Summary of reasoning: Among the given matrices, only the first matrix yields a nonzero determinant, which is a necessary and sufficient condition for invertibility in the 2x2 case. All other options produce a determinant of zero, signifying non-invertibility.

Which of the following matrices is invertible?单项选择题

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